Precalculus Examples

$f (x) = \frac{\sqrt{x^{2} - 1}}{x}$ , $1 \leq x \leq 7$

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.

$\frac{f (7) - f (1)}{(7) - (1)}$

Step 2.2

Substitute the equation $y = \frac{\sqrt{x^{2} - 1}}{x}$ for $f (7)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{\sqrt{{(7)}^{2} - 1}}{7}) - (\frac{\sqrt{{(1)}^{2} - 1}}{1})}{(7) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

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

$\frac{\frac{\sqrt{{(7)}^{2} - 1}}{7} - \sqrt{{(1)}^{2} - 1}}{(7) - (1)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $7$ .

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Step 3.2.1

Multiply $\frac{\frac{\sqrt{7^{2} - 1}}{7} - \sqrt{1^{2} - 1}}{7 - (1)}$ by $\frac{7}{7}$ .

$\frac{7}{7} \cdot \frac{\frac{\sqrt{7^{2} - 1}}{7} - \sqrt{1^{2} - 1}}{7 - (1)}$

Step 3.2.2

Combine.

$\frac{7 (\frac{\sqrt{7^{2} - 1}}{7} - \sqrt{1^{2} - 1})}{7 (7 - (1))}$

Step 3.3

Apply the distributive property.

$\frac{7 \frac{\sqrt{7^{2} - 1}}{7} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.4

Cancel the common factor of $7$ .

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Step 3.4.1

Cancel the common factor.

$\frac{7 \frac{\sqrt{7^{2} - 1}}{7} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.4.2

Rewrite the expression.

$\frac{\sqrt{7^{2} - 1} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Raise $7$ to the power of $2$ .

$\frac{\sqrt{49 - 1} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.2

Subtract $1$ from $49$ .

$\frac{\sqrt{48} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.3

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 3.5.3.1

Factor $16$ out of $48$ .

$\frac{\sqrt{16 (3)} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.3.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 3} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.4

Pull terms out from under the radical.

$\frac{4 \sqrt{3} + 7 (- \sqrt{1^{2} - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.5

One to any power is one.

$\frac{4 \sqrt{3} + 7 (- \sqrt{1 - 1})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.6

Subtract $1$ from $1$ .

$\frac{4 \sqrt{3} + 7 (- \sqrt{0})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.7

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

$\frac{4 \sqrt{3} + 7 (- \sqrt{0^{2}})}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.8

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

$\frac{4 \sqrt{3} + 7 (- 0)}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.9

Multiply $7 (- 0)$ .

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Step 3.5.9.1

Multiply $- 1$ by $0$ .

$\frac{4 \sqrt{3} + 7 \cdot 0}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.9.2

Multiply $7$ by $0$ .

$\frac{4 \sqrt{3} + 0}{7 \cdot 7 + 7 (- (1))}$

Step 3.5.10

Add $4 \sqrt{3}$ and $0$ .

$\frac{4 \sqrt{3}}{7 \cdot 7 + 7 (- (1))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $7$ by $7$ .

$\frac{4 \sqrt{3}}{49 + 7 (- (1))}$

Step 3.6.2

Multiply $7 (- (1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $1$ .

$\frac{4 \sqrt{3}}{49 + 7 \cdot - 1}$

Step 3.6.2.2

Multiply $7$ by $- 1$ .

$\frac{4 \sqrt{3}}{49 - 7}$

Step 3.6.3

Subtract $7$ from $49$ .

$\frac{4 \sqrt{3}}{42}$

Step 3.7

Cancel the common factor of $4$ and $42$ .

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Step 3.7.1

Factor $2$ out of $4 \sqrt{3}$ .

$\frac{2 (2 \sqrt{3})}{42}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $2$ out of $42$ .

$\frac{2 (2 \sqrt{3})}{2 (21)}$

Step 3.7.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{3})}{2 \cdot 21}$

Step 3.7.2.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{21}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{21}$

Decimal Form:

$0.16495721 \dots$