Determining if a Function is Even, Odd, Both, or Neither

To determine whether a function is even, odd, both, or neither, one must employ specific definitions and tests. Understanding these concepts will enable you to classify any function based on its properties.

Definitions

Even Function

A function f(x) is even if for every x in its domain, f(-x) f(x). Graphically, even functions are symmetric with respect to the y-axis.

Odd Function

A function f(x) is odd if for every x in its domain, f(-x) -f(x). Graphically, odd functions are symmetric with respect to the origin.

Neither

If a function does not satisfy the conditions for being even or odd, it is classified as neither.

Steps to Determine

Check for Evenness

Calculate f(-x). If f(-x) f(x) for all x in the domain, then the function is even.

Check for Oddness

Calculate f(-x). If f(-x) -f(x) for all x in the domain, then the function is odd.

Conclusion

If the function satisfies both conditions, it is both even and odd, which is true only for the zero function f(x) 0. If it satisfies neither condition, then it is neither even nor odd.

Example

Let’s apply these steps to the function f(x) x^2

Calculate f(-x). f(-x) (-x)^2 x^2 f(x) Since f(-x) f(x), f(x) x^2 is even.

Now for g(x) x^3

Calculate g(-x). g(-x) (-x)^3 -x^3 -g(x) Since g(-x) -g(x), g(x) x^3 is odd.

For h(x) x - x^3

Calculate h(-x) h(-x) -x - x^3 h(-x) ≠ h(x) and h(-x) ≠ -h(x) Since neither condition is satisfied, h(x) x - x^3 is neither even nor odd.

Understanding Even and Odd Functions

Even functions are symmetrical about the y-axis.

Odd functions have rotational symmetry about the origin. This means if you ROTATE them 180 degrees, they look just the same.

Even functions: y x^2, y cosx Odd functions: y x^3, y sinx Neither: y x - x^2 - x^3

Example of Even and Odd Functions Graphs

Even function: y x^2 The graph of y x^2 is symmetric about the y-axis. Odd function: y x^3 The graph of y x^3 is not symmetric about the y-axis but is symmetric about the origin.

Determining if a Function is Even or Odd

For a polynomial function f(x), if all the powers of x are even, it is even. If all the powers of x are odd, it is odd. If the powers of x are not all of the same parity, the function is neither even nor odd.

Conclusion

A function is even if its graph is symmetric about the y-axis, plugging -x into the function results in the exact same function, or all of its exponents are even (applies only to polynomials).

A function is odd if its graph is symmetric about the origin, plugging -x into the function results in the negative value of the function, or all of its exponents are odd (applies only to polynomials).

Remember, a function can also be neither even nor odd, satisfying none of the conditions mentioned above.