determine whether y is a function of x calculator
For this purpose, it substitutes x in the given function \( f (x) = 2x^2 3 \) and then simplifies. Press the "0" button. For example, the black dots on the graph in the graph belowtell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1[/latex]. Hence, Directly proportional variables are indicated graphically by a straight line passing through the origin of the coordinate plane. An additional way to consider the absolute value would be to examine a number line with zero in the center. Calculated Industries 3405 Real Estate Master IIIX finance calculator; protective slide cover; quick reference guide; pocket user's guide; batteries. This means that execution reaches the end of your function and doesn't see a return statement, so it returns None. https://www.desmos.com/calculator/dcq8twow2q, http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant, [latex]f\left(x\right)=\frac{1}{x}[/latex], [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex], [latex]f\left(x\right)=\sqrt[3]{x}[/latex], Verify a function using the vertical line test, Verify a one-to-one function with the horizontal line test, Identify the graphs of the toolkit functions. The quadratic function y = x 2 - x - 2 is plotted below:. We discuss preparing a mortgage loan payment calculator using the PMT formula, practical examples, and a downloadable Excel template. (a,b) > 0, then f has a relative minimum at (a,b). If no vertical line can intersect the curve more than once, the graph does represent a function. A vertical line includes all points with a particular [latex]x[/latex] value. Justify your answer. It will discover the sine of 8, the preceding outcome. Some of these functions are programmed to individual buttons on many calculators. Using a graphing calculator is a fast and effective way to identify the X and Y intercepts of a function. The composition of an even and an odd function is even. The function in (b) is one-to-one. Press the "Zoom" button. A: To determine whether the equation given below :- x2+y=25- (i) defines y as a function of x Q: Determine whether the equation defines y as a function of x : x + y3 = 8 A: Given: x+y3=8 for finding given equation as a function of x, we find value of y in terms of x so, Q: Determine whether the equation defines y as a function of x. y = 3 x Non-integer numbers like 3.462, 7/9, or infinity are neither odd nor even. A mortgage calculator is used to compute the value of the monthly installment payable by the borrower on the mortgage loan. Enter the equation paying careful attention to include all parentheses and operators. The device measures 5.75 x 3 x 0.6 inches and weighs 3.2 ounces, and it's backed by a one-year limited warranty. Remember that a linear relationship does not necessarily indicate a directly proportional relationship. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. In this case the result is computed in the function add2_New() and no print statement, and returned to the caller who then prints it in turn. The table below shows the signs of D and fxx(a,b). Cox holds a Bachelor of Science in economics with a minor in international business from the University of Kentucky. PMT function includes three mandatory and two optional parameters. Editorial Values. f (- x) f (x) not an even function. Copyright 2022 . How To Calculate Uncertainty With The Easy Calculator Determine whether y is a function of calculator review key you may use no necessary transformation fx 115es 2 solving for or f x using scientific view as find the value and how to operate sketch graph chegg 1 four ways represent Trending Posts I Love You Math Equation Gif Explore math with our beautiful, free online graphing calculator. In both calendar systems, February is the only month with a variable amount of days, it has 29 days during a leap year, and 28 days during all other years. Python: To check whether a given year is a leap year or not? fx(x,y) = 4x - 4y Which of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex]. Use it to try out great new products and services nationwide without paying full pricewine, food delivery, clothing and more. $$ f (- x) = (- x)^3 (- x)^2 1 $$ Example: We now determine the second order partial derivatives. fx(x,y) = 0 to find the critical points.. read more of the loan. 3405 Real Estate Master IIIX Finance Calculator At a Glance: Only RFID Journal provides you with the latest insights into whats happening with the technology and standards and inside the operations of leading early adopters across all industries and around the world. Example: If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/a3\/Determine-Whether-Two-Variables-Are-Directly-Proportional-Step-1-Version-2.jpg\/v4-460px-Determine-Whether-Two-Variables-Are-Directly-Proportional-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/a\/a3\/Determine-Whether-Two-Variables-Are-Directly-Proportional-Step-1-Version-2.jpg\/aid391887-v4-728px-Determine-Whether-Two-Variables-Are-Directly-Proportional-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"
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\n<\/p><\/div>"}. The first line comes form the print statement inside of add2(). He brings expertise in energy, home improvement, business and electronics. This article has been viewed 304,762 times. $$ f (x) = x^3 x^2 1 $$ If you ignore this default value will be zero. We now use the equation y = x3 to find the critical points. Sin, cos, and tan are trigonometric functions, they can be expressed as odd or even functions as well. A vertical line includes all points with a particular [latex]x[/latex] value. Mathematically, we can define it as How do I check whether a file exists without exceptions? We use cookies to make wikiHow great. Search: Determine Whether Y Is A Function Of X Calculator. $$ f (x) = x^3 x^2 1 $$ If the year is a leap year this will result in True (or 1, if used in C), otherwise it will return False (or 0, if used in C). Cos (-x) = cos x Calculate Your Body Mass Index. Zero is an integer multiply of 2 such as 0 x 2, due to this reason we can ask zero is an even number. The sets of odd and even number can be represented as: A formal definition of an odd number is an integer of the form n = 2x + 1, where x is an integer. Which is not an odd function. function every(f) // Returns true if every element in the list satisfies the predicate f. Returns true for an empty list. buffer_radius: DOUBLE that represents the radius of the buffer around the input geography. Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the latest update on a major storm, Sciencing is here to help. That means it's not a function let x=1, then y = -1 or +1, if there is not a unique value of y for any value of x, then it's not a function. You can download this Mortgage Calculator Excel Template here . In this instance, all terms require a denominator of 4x. The x-coordinate of the first point is 2, and the x-coordinate of the second point is 4. If there is any such line, the graph does not represent a function. Now, add x in the given function. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. By using our website, you agree to our use of cookies (, Mortgage Payment Loan Calculator in Excel, Formula to Calculate Mortgage Payment in Excel. If you can use algebra to rewrite the equation to read y=kx, wherein y = the y coordinate, k = the constant, and x = the x coordinate, then the variables are directly proportional. $$ f (x) f (-x) = f (x) $$ How does the Chameleon's Arcane/Divine focus interact with magic item crafting? Some equations have more than one X intercept; perform these steps for each one. $$ f (- x) = (x^7 + 8 x^5 x^3 + 6x) $$ Tan (-x) = tan x It is a "direct variation," because each x value is related identically to the corresponding y value (x = 2y). Factor and solve for y. This value is compared against 0 with the less-than operator. A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of The Future Value (FV) formula is a financial terminology used to calculate cash flow value at a futuristic date compared to the original receipt. Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. The graph of a quadratic function is a parabola, a 2-dimensional curve that looks like either a cup () or a cap (). When two variables are directly proportional, they change at the same rate. Take any function To compute the x intercept set y = 0 and solve the function Set x = 0 to get the y intercept value. fy(x,y) = - 4y3 + 4x If plugin the (- x) into the function f (x) and we dont get either even or odd, then that implies the given function f (x) is neither odd nor even function. $$ f (- x) = (- 1) x^7 + 8 (- 1) x^5 (- 1) x^3 6 (x) $$ If you are diagnosed with high blood pressure, your healthcare provider may advise you to take one or more medicines to keep it under control. $$ f (- x) f (x) And f (- x) f (x) $$ Now, substitute x into the given function f (x) and simplifying, Once you understand these basic concepts, it is easy to identify directly proportional variables by using the equation of their line, or their values. * Please provide your correct email id. if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[580,400],'analyzemath_com-box-4','ezslot_4',260,'0','0'])};__ez_fad_position('div-gpt-ad-analyzemath_com-box-4-0'); Solution to Example 2: The general form of a quadratic function is: f (x) = ax2 + bx + c (or y = ax2 + bx + c) , where a, b and c are all real numbers and a cannot be equal to 0. Example: Use of a boolean expression to determine whether or not a transition is taken. The transition from the Julian to Gregorian calendar system occurred in 1918, when the next day after January 31st was February 14th. % of people told us that this article helped them. For example. For each of the following equations, determine whether \ ( y \) is a function of \ ( x \). The [latex]x[/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[/latex] value. In this case the coefficient is -0.541 meaning that there exists a moderate inverse association between X and Y. The easiest way to determine your BMR (Basal Metabolic Rate) is by using the Mifflin-St Jeor formula. Algebra questions and answers. Determine whether a given point is on a line, given the equation of the line Linear Equations Determine the vertex, axis of symmetry, focus and directrix of a parabola, given its equation. If a function does not express symmetry, then the function can be neither odd nor even. The missing part is the use of return statement: I tried solving in this way and it as work for me like a charm !!! Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Compare the factor, or constant, of the two variables. For example: The function f (x) = x^2 The .gov means it's official. Does a 120cc engine burn 120cc of fuel a minute? An online even odd or neither calculator determine whether the function is odd, even, or neither by the following steps: Cosine is an even function and sin is an odd function. The objective of the FV equation is to determine the future value of a prospective investment and whether the returns yield sufficient returns to factor in the time value of money. (a) y = 4 x 3 so it (b) y = 2 x 2 3 so it (c) y = 6 x 2 3 so it Graph the equation. Graph each toolkit function using function notation. Note: This does not address any possible problems you have with your leap Now, look at the graph of a function f (x) = x^3. source. Sin (-x) = -sin x. fxy(x,y) = 4 When too many critical points are found, the use of a table is very convenient. Therefore, the variables are directly proportional. What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked. The vertical line test can be used to determine whether a graph represents a function. As a small thank you, wed like to offer you a $30 gift card (valid at GoNift.com). Hence. How do I execute a program or call a system command? Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. Given function: If the year is evenly divisible by 100, this will result in a value of 1, otherwise it will result in a value of 0. We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators. Determine the y-intercept. Include your email address to get a message when this question is answered. In linear algebra, Cramers rule is a specific formula used for solving a system of linear equations containing as many equations as unknowns, efficient whenever the system of equations has a unique solution. A function f (x) is odd, when f (- x) = f (x), for all x in the given function. Which has the solutions. Hence, f (- x) = f (x), which means if we substitute the same values in an online even or odd calculator, it displays the same results that are even function. So potentially, the variables are directly proportional. Please note: this code is horribly inefficient, incredibly unreadable, and a detriment to any code attempting to follow proper practices. $$ f (x) f (-x) = f (x) $$ eslint: generator-star-spacing. Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. This rule is named after Gabriel Cramer (17041752), who published the rule for an arbitrary number of unknowns in 1750. Why? Graphs display many input-output pairs in a small space. You may be asked to find algebraically whether the function is odd, even, or neither. The graph of a quadratic function is a parabola that opens up or down If you take a look on the function graphs, you see that intersects the y-axis at intersects the y-axis at Solution: Well, you can use an online odd or even function calculator to check whether a function is even, odd or neither P.o. How do I tell if this single climbing rope is still safe for use? Evan Cox has been working in the energy industry since 2007. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Disable your Adblocker and refresh your web page . By treating high blood pressure, you reduce your risk of heart disease, stroke, kidney failure and eye damage. x is the input, y is the output for an input of 1, there are 2 possible outputs, 1 function and * are part of the same conceptual keyword - * is not a modifier for function, function* is a This article was co-authored by wikiHow Staff. Example: You are free to use this image on your website, templates, etc., Please provide us with an attribution link. A horizontal line includes all points with a particular [latex]y[/latex] value. $$ f (x) = 2x^2 3 $$ Well, you can use an online odd or even function calculator to check whether a function is even, odd or neither. It's similar to the Mark's answer, but short circuits at the first test (note the parenthesis). When working with functions, it is similarly helpful to have a base set of building-block elements. $$ f (- x) = (x^3 + x^2 + 1) $$ Federal government websites often end in .gov or .mil. Did you have an idea for improving this content? The graphs and sample table values are included with each function shown below. Now, plug in x in the function, The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. not a function 15 - x 2 y = * V5- x 2 ; eg . Now for the future, we can change the values and have our mortgage calculator in Excel. Spherical harmonics can represent any scalar field (function of position) that satisfies certain properties. Like many other excel mortgage calculatorMortgage CalculatorA mortgage calculator is used to compute the value of the monthly installment payable by the borrower on the mortgage loan. Therefore, an online even odd or neither calculator is able to determine whether a function is odd or even. Alternatively, you can use the standard library's calendar.isleap, which has exactly the same implementation: So 1900 doesn't enter the if clause because 1900 % 100 != 0 is False, But 1900 also doesn't enter the else clause because 1900 % 4 != 0 is also False. Un-Answered Problems With Solve for X Calculator Exposed Get the Scoop on Solve for X Calculator Before You're Too Late . To use the function calculator, follow these steps: Step 1: Enter the function f (x) in the appropriate input field Step 2: Click on Graph to get the results Step 3: The graph of the function will appear in a new window Function Calculator Function Notation And Calculator Explanation & Examples Now to calculate your TDEE, simply multiply your BMR by your activity level. Enter the equation. Scroll down to "Zero" and press "Enter." fyy(x,y) = 12y2 So, In this exercise, you will graph the toolkit functions using an online graphing tool. The odd or even function calculator displays the nature of the function as even, odd, or neither. $$ f (- x) = 2(- 1x)^2 3 $$ In the Julian calendar, leap years are divisible by 4 while in the Gregorian calendar, leap years are either of the following: To subscribe to this RSS feed, copy and paste this URL into your RSS reader. By using our site, you agree to our. This function is similar to ST_BUFFER, but you provide tolerance instead of segments to determine how much the resulting geography can deviate from the ideal buffer radius. In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. It will need to include the X and Y intercepts. By the fundamental theorem of calculus, it can be seen that the integral of the acceleration function a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to the change of velocity. $$ f (x) = x^7 + 8x^5 x^3 + 6x $$ fy(x,y) = - 4x + 4y3 How to Use a Function Calculator? Solution: However, an online Composite Function Calculator can help you to evaluate the composition of the functions from entered values of functions f(x) and g(x) at the specific points. Solution to Example 3: So he approached the bank, and the bank agreed to sanction the loan based on the below conditions. The higher the number of cigarettes, the lower the longevity - a dose-dependent relationship. Add this calculator to your site and lets users to perform easy calculations. Find centralized, trusted content and collaborate around the technologies you use most. Is it possible to hide or delete the new Toolbar in 13.1? Feel free to contact us at your convenience! We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. $$ f (- x) = (- 1) x^3 (1) x^2 1 $$ This means that 32nd day in 1918, was the February 14th. Incase you find any issue in my code feel free to guide me. Not the answer you're looking for? If it is divisible by both 100 and 400, it will result in 0. Both members and non-members can engage with resources to support the implementation of the Notice and Wonder strategy on this webpage. We now determine the second order partial derivatives. $$ f (- x) = f (x) $$ Download Mortgage Calculator Excel Template, Introduction to Excel, Excel Basic and Advanced Functions and others. $$ f (- x) = x^3 x^2 1 $$ $$ f (- x) = 2 (- 1)^2 (x)^2 3 $$ In this article, we will look at the definitions, properties, and how to find if a function is even or odd. If you can use algebra to rewrite the equation to read y=kx, wherein y = the y coordinate, k = the constant, and x = the x coordinate, then the variables are directly proportional. If you're referring to inverse proportionality, such a case can be represented by xy = k, or y = k/x, or x = k/y, where x and y are variables, and k is a constant. Graphs of Functions, Equations, and Algebra, The Applications of Mathematics It means that the function f (x) is the same for the positive and negative x-axis, or graphically, symmetric about the y-axis: Here, How to determine a Python variable's type? In your example of ordered pairs x = 2, 4, 6 and y = 1, 2, 3, only the first two a and y values increased by the same constant of 2, yet you said that the example was a direct variation. Before sharing sensitive information, make sure you're on a federal government site. geography: The input GEOGRAPHY to encircle with the buffer radius. Justify your answer. As we have seen in examples above, we can represent a function using a graph. Disconnect vertical tab connector from PCB. Combined with the second equation, we obtain. 4y(-1 + y2) = 0 Using a graphing calculator is a fast and effective way to identify the X and Y intercepts of a function. Note: This does not address any possible problems you have with your leap year computation, but ANSWERS YOUR SPECIFIC QUESTION as to why you are getting None as a result of your function call in conjunction with your print. Press the "Trace" button. Consider the functions (a), and (b)shown inthe graphsbelow. Mr. A wants to buy a car, which costs 600,000. All common integration techniques and even special functions are supported. This type of classification applies only to integers. A function has only one output value for each input value. Espaol. fx(x,y) = - 4x3 + 4y in Physics and Engineering, Exercises de Mathematiques Utilisant les Applets, Trigonometry Tutorials and Problems for Self Tests, Elementary Statistics and Probability Tutorials and Problems, Free Practice for SAT, ACT and Compass Math tests, Optimization Problems with Functions of Two Variables, Free Mathematics Tutorials, Problems and Worksheets (with applets), Critical Points of Functions of Two Variables. While looking at the function that needed to be graphed for assignment, a student or tutor can recognize with our calculator that would go quickly because the signed values do not matter in the calculations of function values. For Odd Function: Mathematics College answered expert verified For each of the following equations, determine whether y is a function of x. x= -7y^2 b. x = 1/4y c. y^2 = 4x +9 d. y= 5x +8 1 See answer Advertisement sqdancefan 9514 1404 393 Answer: a) not a function b) is a function c) not a function d) is a function Step-by-step explanation: It considers the loan amount, the annual rate of interest, and the repayment frequency for calculation. The logic in the "one-liner" works fine. $$ f (- x) = 2x^2 3 $$ linear function, an exponential function, or a quadratic function In other words, y is a function of x . Help us identify new roles for community members, Proposing a Community-Specific Closure Reason for non-English content, I tried to make list comprehension for leap year identifier code in python, How to find the next leap year since certain year, numpy datetime64 add or substract date interval, Whey is_leap_year returning None instead of True. Go through these steps and understand them to compute the function easily. 2022 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. Our calculator was developed with the assistance of fitness professionals using a variety of equations. Our goal is to make science relevant and fun for everyone. Finally, 1 is subtracted from this remaining value, resulting in -1 if the year is a leap year, and either 0, 1, or 2 if it is not. All tip submissions are carefully reviewed before being published. Caution: The results for this test can be misleading unless you have made a scatter plot first to ensure your data roughly fits a straight line. Revised annually, the latest version contains employment projections for the 2021-31 decade. Determine whether the equation defines y as a function of x Thank you so much for your help!! A function f(x) is even, when f (x) = f (-x) for all values of x. To do this, we divide both sides by 2. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable. So, to clear the loan of 450,000 in 3 years at an interest rate of 15%, Mr. A has to pay 15,599 per month. Find the Y intercept. This will move the cursor to the Y intercept where X = 0. We now use the equation x = y to find the critical points. If you use an even or odd function calculator, then it shows the obtained function is not a starting function. that is also 1:1 compatible with C. (y % 4) >>>It first checks if the year is a leap year via the typical mod-4 check. No. It will also show you the extra amount you are paying as a percentage. How to tell about even odd or neither functions, when: Does a direct proportion need to have a slope of 0? If there is any such line, the function is not one-to-one. I am pretty sure that I should get 1900 is a leap year. Since then they we use the Gregorian calendar system. Your function doesn't return anything, so that's why when you use it with the print statement you get None.So either just call your function like this: leapyr(1900) or modify your function to return a value (by using the return statement), which then would be printed by your print statement.. More on Optimization Problems with Functions of Two Variables in this web site. x(x4 - 1)(x4 + 1) = 0 The objective of the FV equation is to determine the future value of a prospective investment and whether the returns yield sufficient returns to factor in the time value of money. If you don't want to import calendar and apply .isleap method you can try this: In the Gregorian calendar, three conditions are used to identify leap years: This means that in the Gregorian calendar, the years 2000 and 2400 are leap years, while 1800, 1900, 2100, 2200, 2300 and 2500 are NOT leap years. The [latex]y[/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[/latex] value. (Note that, as in the other answer, you have to return something rather than print it.). For example, if the first x-coordinate is 2, and the second x-coordinate is 4, you need to determine what you multiply 2 by to get 4: For example, if the first y-coordinate is 1, and the second y-coordinate is 2, you need to determine what you multiply 1 by to get 2: For example, since the x-coordinates changed by a factor of 2 while the y-coordinates also changed by a factor of 2, the two variables are directly proportional. Select a zoom that will work for your equation. Remember that if one variable rises with the other, it does not necessarily mean that they are directly proportional. From the source of Lumen Learning: Determine whether a function is even, odd, or neither from its graph, Set Representation of Even and Odd Numbers, Properties of Even and Odd Numbers, Even and Odd Decimals. Further, be aware that ZeroDivisionErrors are a consequence of the input year equaling 0, and must be accounted for. The Bureau of Labor Statistics is the principal fact-finding agency for the Federal Government in the broad field of labor economics and statistics. $$ f (- x) = 2(- x)^2 3 $$ - 4x + 4y3 = 0 ((((y % 4) + (int((y - (y % 100)) / y) * ((y % 400) / 100))) - 1) < 0). When learning to read, we start with the alphabet. Therefore, the function f (x) is neither odd nor even. Find the X intercept(s). From the source of Libre Text: Odd and Even Functions, Types of Functions: Even, Odd, or Neither, Set Representation of Even and Odd Numbers, Neither Odd nor Even. Effect of coal and natural gas burning on particulate matter pollution, Allow non-GPL plugins in a GPL main program, I want to be able to quit Finder but can't edit Finder's Info.plist after disabling SIP. About Our Coalition. Determine the critical points by solving the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Connect and share knowledge within a single location that is structured and easy to search. ((y % 400) / 100))) >>>Next, the year is divided by 400 (and subsequently 100, to return 1, 2, or 3 if it is not. The whole formula can be contained in a single expression: Your function doesn't return anything, so that's why when you use it with the print statement you get None. x = 0 , -1 and 1. To tell if 2 variables are directly proportional while looking at their graphs, note if their x- and y- intercept coordinates are changed by the same factor. Scroll to the right of the intercept and press "Enter'" twice. This article is a guide to Mortgage Calculator in Excel. Long answer: Lets say we have the equation x = 2y. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. fyy(x,y) = -12y2 In other words, y is a function of x. By definition, a leap year is divisible by four, but not by one hundred, unless it is divisible by four hundred. If we put (- x) into the function f(x) and get the starting or original function again, then this shows that f(x) is an even function. 