Precalculus Examples

$g (x) = \frac{3}{1 + x^{2}}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$g (- x) = \frac{3}{1 + {(- x)}^{2}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Apply the product rule to $- x$ .

$g (- x) = \frac{3}{1 + {(- 1)}^{2} x^{2}}$

Step 1.2.2

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

$g (- x) = \frac{3}{1 + 1 x^{2}}$

Step 1.2.3

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

$g (- x) = \frac{3}{1 + x^{2}}$

Step 2

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

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Step 2.1

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

Step 2.2

Since $\frac{3}{1 + x^{2}} = \frac{3}{1 + x^{2}}$ , the function is even.

The function is even

Step 3