Precalculus Examples

$f (x) = \frac{\sqrt{x^{2} - 1}}{x}$ , $1 \leq x \leq 3$

Step 1

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

$y = \frac{\sqrt{x^{2} - 1}}{x}$

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.

$\frac{f (3) - f (1)}{(3) - (1)}$

Step 2.2

Substitute the equation $y = \frac{\sqrt{x^{2} - 1}}{x}$ for $f (3)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{\sqrt{{(3)}^{2} - 1}}{3}) - (\frac{\sqrt{{(1)}^{2} - 1}}{1})}{(3) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Divide $\sqrt{{(1)}^{2} - 1}$ by $1$ .

$\frac{\frac{\sqrt{{(3)}^{2} - 1}}{3} - \sqrt{{(1)}^{2} - 1}}{(3) - (1)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.2.1

Multiply $\frac{\frac{\sqrt{3^{2} - 1}}{3} - \sqrt{1^{2} - 1}}{3 - (1)}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\frac{\sqrt{3^{2} - 1}}{3} - \sqrt{1^{2} - 1}}{3 - (1)}$

Step 3.2.2

Combine.

$\frac{3 (\frac{\sqrt{3^{2} - 1}}{3} - \sqrt{1^{2} - 1})}{3 (3 - (1))}$

Step 3.3

Apply the distributive property.

$\frac{3 \frac{\sqrt{3^{2} - 1}}{3} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

$\frac{3 \frac{\sqrt{3^{2} - 1}}{3} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.4.2

Rewrite the expression.

$\frac{\sqrt{3^{2} - 1} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 - 1} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.2

Subtract $1$ from $9$ .

$\frac{\sqrt{8} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.5.3.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.3.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 2} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.4

Pull terms out from under the radical.

$\frac{2 \sqrt{2} + 3 (- \sqrt{1^{2} - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.5

One to any power is one.

$\frac{2 \sqrt{2} + 3 (- \sqrt{1 - 1})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.6

Subtract $1$ from $1$ .

$\frac{2 \sqrt{2} + 3 (- \sqrt{0})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.7

Rewrite $0$ as $0^{2}$ .

$\frac{2 \sqrt{2} + 3 (- \sqrt{0^{2}})}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 \sqrt{2} + 3 (- 0)}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.9

Multiply $3 (- 0)$ .

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Step 3.5.9.1

Multiply $- 1$ by $0$ .

$\frac{2 \sqrt{2} + 3 \cdot 0}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.9.2

Multiply $3$ by $0$ .

$\frac{2 \sqrt{2} + 0}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.10

Add $2 \sqrt{2}$ and $0$ .

$\frac{2 \sqrt{2}}{3 \cdot 3 + 3 (- (1))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $3$ by $3$ .

$\frac{2 \sqrt{2}}{9 + 3 (- (1))}$

Step 3.6.2

Multiply $3 (- (1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $1$ .

$\frac{2 \sqrt{2}}{9 + 3 \cdot - 1}$

Step 3.6.2.2

Multiply $3$ by $- 1$ .

$\frac{2 \sqrt{2}}{9 - 3}$

Step 3.6.3

Subtract $3$ from $9$ .

$\frac{2 \sqrt{2}}{6}$

Step 3.7

Cancel the common factor of $2$ and $6$ .

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Step 3.7.1

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{2 (\sqrt{2})}{6}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $2$ out of $6$ .

$\frac{2 \sqrt{2}}{2 \cdot 3}$

Step 3.7.2.2

Cancel the common factor.

$\frac{2 \sqrt{2}}{2 \cdot 3}$

Step 3.7.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{3}$

Decimal Form:

$0.47140452 \dots$