Precalculus Examples

$f (x) = \frac{1}{\sqrt{x}}$ , $[1, 9]$

Step 1

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

$y = \frac{1}{\sqrt{x}}$

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.

$\frac{f (9) - f (1)}{(9) - (1)}$

Step 2.2

Substitute the equation $y = \frac{1}{\sqrt{x}}$ for $f (9)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{\sqrt{9}}) - (\frac{1}{\sqrt{1}})}{(9) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\sqrt{9} \sqrt{1}$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{1}}}{9 - (1)}$ by $\frac{\sqrt{9} \sqrt{1}}{\sqrt{9} \sqrt{1}}$ .

$\frac{\sqrt{9} \sqrt{1}}{\sqrt{9} \sqrt{1}} \cdot \frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{1}}}{9 - (1)}$

Step 3.1.2

Combine.

$\frac{\sqrt{9} \sqrt{1} (\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{1}})}{\sqrt{9} \sqrt{1} (9 - (1))}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $\sqrt{9}$ .

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Step 3.3.1.1

Factor $\sqrt{9}$ out of $\sqrt{9} \sqrt{1}$ .

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

Step 3.3.1.2

Cancel the common factor.

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

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

Cancel the common factor of $\sqrt{1}$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{\sqrt{1}}$ into the numerator.

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

Step 3.3.2.2

Factor $\sqrt{1}$ out of $\sqrt{9} \sqrt{1}$ .

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

Step 3.3.2.3

Cancel the common factor.

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

Step 3.3.2.4

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Any root of $1$ is $1$ .

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

Step 3.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $3$ by $- 1$ .

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

Step 3.4.5

Subtract $3$ from $1$ .

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.5.2

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

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

Step 3.5.3

Any root of $1$ is $1$ .

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

Step 3.5.4

Multiply $3 \cdot 1 \cdot 9$ .

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Step 3.5.4.1

Multiply $3$ by $1$ .

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

Step 3.5.4.2

Multiply $3$ by $9$ .

$\frac{- 2}{27 + \sqrt{9} \sqrt{1} (- (1))}$

Step 3.5.5

Rewrite $9$ as $3^{2}$ .

$\frac{- 2}{27 + \sqrt{3^{2}} \sqrt{1} (- (1))}$

Step 3.5.6

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

$\frac{- 2}{27 + 3 \sqrt{1} (- (1))}$

Step 3.5.7

Any root of $1$ is $1$ .

$\frac{- 2}{27 + 3 \cdot 1 (- (1))}$

Step 3.5.8

Multiply $3 \cdot 1 (- (1))$ .

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Step 3.5.8.1

Multiply $3$ by $1$ .

$\frac{- 2}{27 + 3 (- (1))}$

Step 3.5.8.2

Multiply $- 1$ by $1$ .

$\frac{- 2}{27 + 3 \cdot - 1}$

Step 3.5.8.3

Multiply $3$ by $- 1$ .

$\frac{- 2}{27 - 3}$

Step 3.5.9

Subtract $3$ from $27$ .

$\frac{- 2}{24}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Cancel the common factor of $- 2$ and $24$ .

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Step 3.6.1.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{24}$

Step 3.6.1.2

Cancel the common factors.

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Step 3.6.1.2.1

Factor $2$ out of $24$ .

$\frac{2 \cdot - 1}{2 \cdot 12}$

Step 3.6.1.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 12}$

Step 3.6.1.2.3

Rewrite the expression.

$\frac{- 1}{12}$

Step 3.6.2

Move the negative in front of the fraction.

$- \frac{1}{12}$