Precalculus Examples

$f (x) = \frac{4}{x^{2} + 1}$ ; $[- 1, 1]$

Step 1

Write $f (x) = \frac{4}{x^{2} + 1}$ as an equation.

$y = \frac{4}{x^{2} + 1}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (1) - f (- 1)}{(1) - (- 1)}$

Step 2.2

Substitute the equation $y = \frac{4}{x^{2} + 1}$ for $f (1)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{4}{{(1)}^{2} + 1}) - (\frac{4}{{(- 1)}^{2} + 1})}{(1) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(1^{2} + 1) ({(- 1)}^{2} + 1)$ .

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Step 3.1.1

Multiply $\frac{\frac{4}{1^{2} + 1} - \frac{4}{{(- 1)}^{2} + 1}}{1 - (- 1)}$ by $\frac{(1^{2} + 1) ({(- 1)}^{2} + 1)}{(1^{2} + 1) ({(- 1)}^{2} + 1)}$ .

$\frac{(1^{2} + 1) ({(- 1)}^{2} + 1)}{(1^{2} + 1) ({(- 1)}^{2} + 1)} \cdot \frac{\frac{4}{1^{2} + 1} - \frac{4}{{(- 1)}^{2} + 1}}{1 - (- 1)}$

Step 3.1.2

Combine.

$\frac{(1^{2} + 1) ({(- 1)}^{2} + 1) (\frac{4}{1^{2} + 1} - \frac{4}{{(- 1)}^{2} + 1})}{(1^{2} + 1) ({(- 1)}^{2} + 1) (1 - (- 1))}$

Step 3.2

Apply the distributive property.

$\frac{(1^{2} + 1) ({(- 1)}^{2} + 1) \frac{4}{1^{2} + 1} + (1^{2} + 1) ({(- 1)}^{2} + 1) (- \frac{4}{{(- 1)}^{2} + 1})}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $1^{2} + 1$ .

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Step 3.3.1.1

Cancel the common factor.

Step 3.3.1.2

Rewrite the expression.

$\frac{({(- 1)}^{2} + 1) \cdot 4 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- \frac{4}{{(- 1)}^{2} + 1})}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.3.2

Cancel the common factor of ${(- 1)}^{2} + 1$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{4}{{(- 1)}^{2} + 1}$ into the numerator.

$\frac{({(- 1)}^{2} + 1) \cdot 4 + (1^{2} + 1) ({(- 1)}^{2} + 1) \frac{- 4}{{(- 1)}^{2} + 1}}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.3.2.2

Factor ${(- 1)}^{2} + 1$ out of $(1^{2} + 1) ({(- 1)}^{2} + 1)$ .

$\frac{({(- 1)}^{2} + 1) \cdot 4 + ({(- 1)}^{2} + 1) (1^{2} + 1) \frac{- 4}{{(- 1)}^{2} + 1}}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{({(- 1)}^{2} + 1) \cdot 4 + ({(- 1)}^{2} + 1) (1^{2} + 1) \frac{- 4}{{(- 1)}^{2} + 1}}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{({(- 1)}^{2} + 1) \cdot 4 + (1^{2} + 1) \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

$\frac{(1 + 1) \cdot 4 + (1^{2} + 1) \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.2

Add $1$ and $1$ .

$\frac{2 \cdot 4 + (1^{2} + 1) \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.3

Multiply $2$ by $4$ .

$\frac{8 + (1^{2} + 1) \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.4

One to any power is one.

$\frac{8 + (1 + 1) \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.5

Add $1$ and $1$ .

$\frac{8 + 2 \cdot - 4}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.6

Multiply $2$ by $- 4$ .

$\frac{8 - 8}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.4.7

Subtract $8$ from $8$ .

$\frac{0}{(1^{2} + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

One to any power is one.

$\frac{0}{(1 + 1) ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.2

Add $1$ and $1$ .

$\frac{0}{2 ({(- 1)}^{2} + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.3

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

$\frac{0}{2 (1 + 1) \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.4

Add $1$ and $1$ .

$\frac{0}{2 \cdot 2 \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.5

Multiply $2 \cdot 2 \cdot 1$ .

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Step 3.5.5.1

Multiply $2$ by $2$ .

$\frac{0}{4 \cdot 1 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.5.2

Multiply $4$ by $1$ .

$\frac{0}{4 + (1^{2} + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.6

One to any power is one.

$\frac{0}{4 + (1 + 1) ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.7

Add $1$ and $1$ .

$\frac{0}{4 + 2 ({(- 1)}^{2} + 1) (- (- 1))}$

Step 3.5.8

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

$\frac{0}{4 + 2 (1 + 1) (- (- 1))}$

Step 3.5.9

Add $1$ and $1$ .

$\frac{0}{4 + 2 \cdot 2 (- (- 1))}$

Step 3.5.10

Multiply $2 \cdot 2 (- (- 1))$ .

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Step 3.5.10.1

Multiply $2$ by $2$ .

$\frac{0}{4 + 4 (- (- 1))}$

Step 3.5.10.2

Multiply $- 1$ by $- 1$ .

$\frac{0}{4 + 4 \cdot 1}$

Step 3.5.10.3

Multiply $4$ by $1$ .

$\frac{0}{4 + 4}$

Step 3.5.11

Add $4$ and $4$ .

$\frac{0}{8}$

Step 3.6

Divide $0$ by $8$ .

$0$