Precalculus Examples

$f (x) = \frac{1}{\sqrt{x}}$ , $[4, 9]$

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.

Tap for more steps...

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.

$\frac{f (9) - f (4)}{(9) - (4)}$

Step 2.2

Substitute the equation $y = \frac{1}{\sqrt{x}}$ for $f (9)$ and $f (4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{\sqrt{9}}) - (\frac{1}{\sqrt{4}})}{(9) - (4)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $\sqrt{9} \sqrt{4}$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{4}}}{9 - (4)}$ by $\frac{\sqrt{9} \sqrt{4}}{\sqrt{9} \sqrt{4}}$ .

$\frac{\sqrt{9} \sqrt{4}}{\sqrt{9} \sqrt{4}} \cdot \frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{4}}}{9 - (4)}$

Step 3.1.2

Combine.

$\frac{\sqrt{9} \sqrt{4} (\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{4}})}{\sqrt{9} \sqrt{4} (9 - (4))}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{9} \sqrt{4} \frac{1}{\sqrt{9}} + \sqrt{9} \sqrt{4} (- \frac{1}{\sqrt{4}})}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3

Simplify by cancelling.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

Factor $\sqrt{9}$ out of $\sqrt{9} \sqrt{4}$ .

$\frac{\sqrt{9} (\sqrt{4}) \frac{1}{\sqrt{9}} + \sqrt{9} \sqrt{4} (- \frac{1}{\sqrt{4}})}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{\sqrt{9} \sqrt{4} \frac{1}{\sqrt{9}} + \sqrt{9} \sqrt{4} (- \frac{1}{\sqrt{4}})}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{\sqrt{4} + \sqrt{9} \sqrt{4} (- \frac{1}{\sqrt{4}})}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.2

Cancel the common factor of $\sqrt{4}$ .

Tap for more steps...

Step 3.3.2.1

Move the leading negative in $- \frac{1}{\sqrt{4}}$ into the numerator.

$\frac{\sqrt{4} + \sqrt{9} \sqrt{4} \frac{- 1}{\sqrt{4}}}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.2.2

Factor $\sqrt{4}$ out of $\sqrt{9} \sqrt{4}$ .

$\frac{\sqrt{4} + \sqrt{4} \sqrt{9} \frac{- 1}{\sqrt{4}}}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{\sqrt{4} + \sqrt{4} \sqrt{9} \frac{- 1}{\sqrt{4}}}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{\sqrt{4} + \sqrt{9} \cdot - 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

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

$\frac{\sqrt{2^{2}} + \sqrt{9} \cdot - 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4.2

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

$\frac{2 + \sqrt{9} \cdot - 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4.3

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

$\frac{2 + \sqrt{3^{2}} \cdot - 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4.4

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

$\frac{2 + 3 \cdot - 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4.5

Multiply $3$ by $- 1$ .

$\frac{2 - 3}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.4.6

Subtract $3$ from $2$ .

$\frac{- 1}{\sqrt{9} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

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

$\frac{- 1}{\sqrt{3^{2}} \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.2

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

$\frac{- 1}{3 \sqrt{4} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.3

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

$\frac{- 1}{3 \sqrt{2^{2}} \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.4

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

$\frac{- 1}{3 \cdot 2 \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.5

Multiply $3 \cdot 2 \cdot 9$ .

Tap for more steps...

Step 3.5.5.1

Multiply $3$ by $2$ .

$\frac{- 1}{6 \cdot 9 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.5.2

Multiply $6$ by $9$ .

$\frac{- 1}{54 + \sqrt{9} \sqrt{4} (- (4))}$

Step 3.5.6

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

$\frac{- 1}{54 + \sqrt{3^{2}} \sqrt{4} (- (4))}$

Step 3.5.7

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

$\frac{- 1}{54 + 3 \sqrt{4} (- (4))}$

Step 3.5.8

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

$\frac{- 1}{54 + 3 \sqrt{2^{2}} (- (4))}$

Step 3.5.9

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

$\frac{- 1}{54 + 3 \cdot 2 (- (4))}$

Step 3.5.10

Multiply $3 \cdot 2 (- (4))$ .

Tap for more steps...

Step 3.5.10.1

Multiply $3$ by $2$ .

$\frac{- 1}{54 + 6 (- (4))}$

Step 3.5.10.2

Multiply $- 1$ by $4$ .

$\frac{- 1}{54 + 6 \cdot - 4}$

Step 3.5.10.3

Multiply $6$ by $- 4$ .

$\frac{- 1}{54 - 24}$

Step 3.5.11

Subtract $24$ from $54$ .

$\frac{- 1}{30}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{1}{30}$