Precalculus Examples

$\frac{x^{2}}{x^{2} + 1}$

Step 1

Write $\frac{x^{2}}{x^{2} + 1}$ as a function.

$f (x) = \frac{x^{2}}{x^{2} + 1}$

Step 2

Find $f (- x)$ .

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Step 2.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \frac{{(- x)}^{2}}{{(- x)}^{2} + 1}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{{(- 1)}^{2} x^{2}}{{(- x)}^{2} + 1}$

Step 2.2.2

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{1 x^{2}}{{(- x)}^{2} + 1}$

Step 2.2.3

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{x^{2}}{{(- x)}^{2} + 1}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Apply the product rule to $- x$ .

$f (- x) = \frac{x^{2}}{{(- 1)}^{2} x^{2} + 1}$

Step 2.3.2

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{x^{2}}{1 x^{2} + 1}$

Step 2.3.3

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{x^{2}}{x^{2} + 1}$

Step 3

A function is even if $f (- x) = f (x)$ .

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Step 3.1

Check if $f (- x) = f (x)$ .

Step 3.2

Since $\frac{x^{2}}{x^{2} + 1} = \frac{x^{2}}{x^{2} + 1}$ , the function is even.

The function is even

Step 4