Content

• Steps
• Method 1 of 2: Algebraic Method
• Method 2 of 2: Graphical method
• Tips
• A warning

Functions can be even, odd, or general (that is, neither even nor odd). The type of function depends on the presence or absence of symmetry. The best way to determine the kind of function is to perform a series of algebraic calculations. But the type of the function can also be found out by its schedule. By learning how to define the kind of functions, you can predict the behavior of certain combinations of functions.

Steps

Method 1 of 2: Algebraic Method

1. 1 Remember what the opposite values ​​of the variables are. In algebra, the opposite value of a variable is written with a “-” (minus) sign. Moreover, this is true for any designation of the independent variable (by the letter ${ displaystyle x}$ or any other letter). If in the original function there is already a negative sign in front of the variable, then its opposite value will be a positive variable. Below are examples of some of the variables and their opposite meanings:
   • The opposite meaning for ${ displaystyle x}$ is an ${ displaystyle -x}$.
   • The opposite meaning for ${ displaystyle q}$ is an ${ displaystyle -q}$.
   • The opposite meaning for ${ displaystyle -w}$ is an ${ displaystyle w}$.
2. 2 Replace the explanatory variable with its opposite value. That is, reverse the sign of the independent variable. For example:
   • ${ displaystyle f (x) = 4x ^ {2} -7}$ turns into ${ displaystyle f (-x) = 4 (-x) ^ {2} -7}$
   • ${ displaystyle g (x) = 5x ^ {5} -2x}$ turns into ${ displaystyle g (-x) = 5 (-x) ^ {5} -2 (-x)}$
   • ${ displaystyle h (x) = 7x ^ {2} + 5x + 3}$ turns into ${ displaystyle h (-x) = 7 (-x) ^ {2} +5 (-x) +3}$.
3. 3 Simplify the new function. At this point, you do not need to substitute specific numeric values ​​for the independent variable. You just need to simplify the new function f (-x) to compare it with the original function f (x). Remember the basic rule of exponentiation: raising a negative variable to an even power will result in a positive variable, and raising a negative variable to an odd power will result in a negative variable.
   • ${ displaystyle f (-x) = 4 (-x) ^ {2} -7}$
     • ${ displaystyle f (-x) = 4x ^ {2} -7}$
   • ${ displaystyle g (-x) = 5 (-x) ^ {5} -2 (-x)}$
     • ${ displaystyle g (-x) = 5 (-x ^ {5}) + 2x}$
     • ${ displaystyle g (-x) = - 5x ^ {5} + 2x}$
   • ${ displaystyle h (-x) = 7 (-x) ^ {2} +5 (-x) +3}$
     • ${ displaystyle h (-x) = 7x ^ {2} -5x + 3}$
4. 4 Compare the two functions. Compare the simplified new function f (-x) with the original function f (x). Write down the corresponding terms of both functions under each other and compare their signs.
   • If the signs of the corresponding terms of both functions coincide, that is, f (x) = f (-x), the original function is even. Example:
     • ${ displaystyle f (x) = 4x ^ {2} -7}$ and ${ displaystyle f (-x) = 4x ^ {2} -7}$.
     • Here the signs of the terms coincide, so the original function is even.
   • If the signs of the corresponding terms of both functions are opposite to each other, that is, f (x) = -f (-x), the original function is even. Example:
     • ${ displaystyle g (x) = 5x ^ {5} -2x}$, but ${ displaystyle g (-x) = - 5x ^ {5} + 2x}$.
     • Note that if you multiply each term in the first function by -1, you get the second function. Thus, the original function g (x) is odd.
   • If the new function does not match any of the above examples, then it is a general function (that is, neither even nor odd). For example:
     • ${ displaystyle h (x) = 7x ^ {2} + 5x + 3}$, but ${ displaystyle h (-x) = 7x ^ {2} -5x + 3}$... The signs of the first terms of both functions are the same, and the signs of the second terms are opposite. Therefore, this function is neither even nor odd.

Method 2 of 2: Graphical method

1. 1 Plot a function graph. To do this, use graph paper or a graphing calculator. Select any multiple of the numeric explanatory variable values ${ displaystyle x}$ and plug them into the function to calculate the values ​​of the dependent variable ${ displaystyle y}$... Draw the found coordinates of the points on the coordinate plane, and then connect these points to build a graph of the function.
   • Substitute positive numeric values ​​into the function ${ displaystyle x}$ and corresponding negative numeric values. For example, given the function ${ displaystyle f (x) = 2x ^ {2} +1}$... Plug in the following values ${ displaystyle x}$:
     • ${ displaystyle f (1) = 2 (1) ^ {2} + 1 = 2 + 1 = 3}$... Got a point with coordinates ${ displaystyle (1,3)}$.
     • ${ displaystyle f (2) = 2 (2) ^ {2} + 1 = 2 (4) + 1 = 8 + 1 = 9}$... Got a point with coordinates ${ displaystyle (2.9)}$.
     • ${ displaystyle f (-1) = 2 (-1) ^ {2} + 1 = 2 + 1 = 3}$... Got a point with coordinates ${ displaystyle (-1,3)}$.
     • ${ displaystyle f (-2) = 2 (-2) ^ {2} + 1 = 2 (4) + 1 = 8 + 1 = 9}$... Got a point with coordinates ${ displaystyle (-2.9)}$.
2. 2 Check if the graph of the function is symmetrical about the y-axis. Symmetry refers to the mirroring of the chart about the ordinate axis. If the portion of the graph to the right of the y-axis (positive explanatory variable) coincides with the portion of the graph to the left of the y-axis (negative values ​​of the explanatory variable), the graph is symmetric about the y-axis. If the function is symmetric about the ordinate, the function is even.
   • You can check the symmetry of the graph by individual points. If the value ${ displaystyle y}$which corresponds to the value ${ displaystyle x}$, matches the value ${ displaystyle y}$which corresponds to the value ${ displaystyle -x}$, the function is even.In our example with the function ${ displaystyle f (x) = 2x ^ {2} +1}$ we got the following coordinates of points:
     • (1.3) and (-1.3)
     • (2.9) and (-2.9)
   • Note that when x = 1 and x = -1 the dependent variable is y = 3, and when x = 2 and x = -2 the dependent variable is y = 9. So the function is even. In fact, in order to find out the exact form of a function, you need to consider more than two points, but the described method is a good approximation.
3. 3 Check if the graph of the function is symmetrical about the origin. The origin is the point with coordinates (0,0). Symmetry about the origin means that a positive value ${ displaystyle y}$ (with a positive value ${ displaystyle x}$) corresponds to a negative value ${ displaystyle y}$ (with a negative value ${ displaystyle x}$), and vice versa. Odd functions are symmetric about the origin.
   • If we substitute several positive and corresponding negative values ​​in the function ${ displaystyle x}$, values ${ displaystyle y}$ will differ in sign. For example, given the function ${ displaystyle f (x) = x ^ {3} + x}$... Substitute multiple values ​​into it ${ displaystyle x}$:
     • ${ displaystyle f (1) = 1 ^ {3} + 1 = 1 + 1 = 2}$... Got a point with coordinates (1,2).
     • ${ displaystyle f (-1) = (- 1) ^ {3} + (- 1) = - 1-1 = -2}$... We got a point with coordinates (-1, -2).
     • ${ displaystyle f (2) = 2 ^ {3} + 2 = 8 + 2 = 10}$... Got a point with coordinates (2,10).
     • ${ displaystyle f (-2) = (- 2) ^ {3} + (- 2) = - 8-2 = -10}$... We got a point with coordinates (-2, -10).
   • Thus, f (x) = -f (-x), that is, the function is odd.
4. 4 Check if the graph of the function has any symmetry. The last type of function is a function whose graph does not have symmetry, that is, there is no mirroring both about the ordinate axis and about the origin. For example, given the function ${ displaystyle f (x) = x ^ {2} + 2x + 1}$.
   • Substitute several positive and corresponding negative values ​​into the function ${ displaystyle x}$:
     • ${ displaystyle f (1) = 1 ^ {2} +2 (1) + 1 = 1 + 2 + 1 = 4}$... Got a point with coordinates (1,4).
     • ${ displaystyle f (-1) = (- 1) ^ {2} +2 (-1) + (- 1) = 1-2-1 = -2}$... We got a point with coordinates (-1, -2).
     • ${ displaystyle f (2) = 2 ^ {2} +2 (2) + 2 = 4 + 4 + 2 = 10}$... Got a point with coordinates (2,10).
     • ${ displaystyle f (-2) = (- 2) ^ {2} +2 (-2) + (- 2) = 4-4-2 = -2}$... We got a point with coordinates (2, -2).
   • According to the results obtained, there is no symmetry. The values ${ displaystyle y}$ for opposite values ${ displaystyle x}$ do not coincide and are not opposite. Thus, the function is neither even nor odd.
   • Note that the function ${ displaystyle f (x) = x ^ {2} + 2x + 1}$ can be written like this: ${ displaystyle f (x) = (x + 1) ^ {2}}$... When written in this form, the function appears to be even because an even exponent is present. But this example proves that the kind of function cannot be quickly determined if the independent variable is enclosed in parentheses. In this case, you need to open the brackets and analyze the resulting exponents.

Tips

• If the exponent of the independent variable is even, then the function is even; if the exponent is odd, the function is odd.

A warning

• This article can only be applied to functions with two variables, the values ​​of which can be plotted on the coordinate plane.