How to Tell if a Function is Even or Odd

Steps for Determining if a Function is Even or Odd

Step 1: Substitute {eq}-x {/eq} for {eq}x {/eq} in {eq}f(x) {/eq}.

Step 2: Compare the result from step 1 with {eq}f(x) {/eq}.

• If the result from step 1 and {eq}f(x) {/eq} are the same, {eq}f(x) {/eq} is an even function.

• If the result from step 1 can be made into {eq}-f(x) {/eq} by factoring out {eq}-1 {/eq}, {eq}f(x) {/eq} is an odd function.

Definitions for Determining if a Function is Even or Odd

Even functions: The function {eq}f {/eq} is an even function if: $$f(-x) = f(x) $$ for all {eq}x {/eq} in the domain of {eq}f {/eq}.

Odd functions: The function {eq}f {/eq} is an odd function if: $$f(-x) = -f(x) $$ for all {eq}x {/eq} in the domain of {eq}f {/eq}.

Let's practice how to tell if a function is even or odd with the following two examples.

Example Problem 1 - How to Tell if a Function is Even or Odd

Determine if the function {eq}p(x) = 3x^6 + x^4 -5x^2 {/eq} is even, odd or neither.

(a) Even, since {eq}p(x) = p(-x) {/eq}

(b) Odd, since {eq}p(x) = p(-x) {/eq}

(c) Odd, since the coefficients of the variables are odd numbers.

(d) Neither, since {eq}p(x) \neq p(-x) {/eq} and {eq}p(-x) \neq -p(x) {/eq}

Step 1: Substitute {eq}-x {/eq} for {eq}x {/eq} in {eq}f(x) {/eq}.

Substituting {eq}-x {/eq} into {eq}p(x) = 3x^6 + x^4 -5x^2 {/eq} gives us: $$\begin{align} p(-x) &= 3(-x)^6 + (-x)^4 -5(-x)^2 \\\\ &= 3x^6 + x^4 - 5x^2 \\\\ \end{align} $$

Step 2: Compare the result from step 1 with {eq}f(x) {/eq}.

• If the result from step 1 and {eq}f(x) {/eq} are the same, {eq}f(x) {/eq} is an even function.

• If the result from step 1 can be made into {eq}-f(x) {/eq} by factoring out {eq}-1 {/eq}, {eq}f(x) {/eq} is an odd function.

Compare {eq}p(x) {/eq} and {eq}p(-x) {/eq}. $$\begin{align} p(x) &= 3x^6 + x^4 - 5x^2 \\ p(-x) &= 3x^6 + x^4 - 5x^2 \end{align} $$ Since {eq}p(x) {/eq} and {eq}p(-x) {/eq} are equivalent, {eq}p(x) {/eq} is an even function.

Choice (a) is the correct answer since {eq}p(x) {/eq} is even because {eq}p(x) = p(-x) {/eq}.

Example Problem 2 - How to Tell if a Function is Even or Odd

Determine if the function {eq}h(x) = 5x^3 - 2x {/eq} is even, odd or neither.

(a) Even, since {eq}h(-x) = -h(x) {/eq}

(b) Odd, since {eq}h(-x) = -h(x) {/eq}

(c) Neither, since the coefficient of {eq}x^3 {/eq}is an odd number and the coefficient of {eq}x {/eq} is an even number.

(d) Neither, since {eq}h(x) \neq h(-x) {/eq} and {eq}h(-x) \neq -h(x) {/eq}

Step 1: Substitute {eq}-x {/eq} for {eq}x {/eq} in {eq}f(x) {/eq}.

Substituting {eq}-x {/eq} into {eq}h(x) = 5x^3 - 2x {/eq} gives us: $$\begin{align} h(-x) &= 5(-x)^3 - 2(-x) \\\\ &= -5x^3 + 2x \\\\ \end{align} $$

Step 2: Compare the result from step 1 with {eq}f(x) {/eq}.

• If the result from step 1 and {eq}f(x) {/eq} are the same, {eq}f(x) {/eq} is an even function.

• If the result from step 1 can be made into {eq}-f(x) {/eq} by factoring out {eq}-1 {/eq}, {eq}f(x) {/eq} is an odd function.

Compare {eq}h(x) {/eq} and {eq}h(-x) {/eq}. $$\begin{align} h(x) &= 5x^3 - 2x \\ h(-x) &= -5x^3 + 2x \end{align} $$

{eq}h(x) {/eq} and {eq}h(-x) {/eq} are the same. Now, we try to factor out {eq}-1 {/eq} from {eq}h(-x) {/eq}. $$\begin{align} h(-x) &= -5x^3 + 2x \\\\ &= -(5x^3 - 2x) \\\\ &=-h(x) \\\\ \end{align} $$

Since {eq}h(-x) = -h(x) {/eq}, the function {eq}h(x) {/eq} is an odd function.

Choice (b) is the correct answer since {eq}h(x) {/eq} is odd because {eq}h(-x) = -h(x) {/eq}.