# How do you determine algebraically if a function is even or odd?

## How do you determine algebraically if a function is even or odd?

In order to “determine algebraically” whether a function is even, odd, or neither, you take the function and plug −x in for x, simplify, and compare the results with what you’d started with.

## How do you prove that a function is even or odd?

To prove a function is odd, plug in x and −x and show they are opposites of each other. −f(x)=−[(x)3]=−x3f(−x)=(−x)3=−x3 So you can see f(−x)=−f(x). Therefore, f(x) is odd.

**What defines an even or odd function?**

Definition. A function f(x) is even if f(-x) = f(x). The function is odd if f(-x) = -f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin.

**How do you show a function is an odd function?**

However, if we evaluate or substitute −x into f ( x ) f\left( x \right) f(x) and get the negative or opposite of the “starting” function, this implies that f ( x ) f\left( x \right) f(x) is an odd function.

### What makes function odd?

A function is odd if −f(x) = f(−x), for all x. The graph of an odd function will be symmetrical about the origin. For example, f(x) = x3 is odd. That is, the function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

### Which of the following functions is an odd function?

Solution : Clearly , g(x)= sin x and `h(x)=log(x+sqrt(x^(2)+1))` both the odd functions. Therefore , f(x)=goh(x) is also an odd functions.

**What makes something an even function?**

A function is an even function if f of x is equal to f of −x for all the values of x. This means that the function is the same for the positive x-axis and the negative x-axis, or graphically, symmetric about the y-axis.

**Is a function even or odd or neither?**

If we get an expression that is equivalent to f(x), we have an even function; if we get an expression that is equivalent to -f(x), we have an odd function; and if neither happens, it is neither!