Precalculus Examples

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

Step 1

Write $f (x) = x^{2} - \frac{1}{x} - 3$ as an equation.

$y = x^{2} - \frac{1}{x} - 3$

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Cancel the common factor of $1$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Combine.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $3$ by $3^{2}$ by adding the exponents.

Tap for more steps...

Step 3.5.1.1

Multiply $3$ by $3^{2}$ .

Tap for more steps...

Step 3.5.1.1.1

Raise $3$ to the power of $1$ .

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

Step 3.5.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.1.2

Add $1$ and $2$ .

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

Step 3.5.2

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

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

Step 3.5.3

Multiply $3$ by $- 3$ .

$\frac{27 - 1 - 9 + 3 (- (1^{2} - 1 \cdot 1 - 3))}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.4

Simplify each term.

Tap for more steps...

Step 3.5.4.1

One to any power is one.

$\frac{27 - 1 - 9 + 3 (- (1 - 1 \cdot 1 - 3))}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.4.2

Multiply $- 1$ by $1$ .

$\frac{27 - 1 - 9 + 3 (- (1 - 1 - 3))}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.5

Subtract $1$ from $1$ .

$\frac{27 - 1 - 9 + 3 (- (0 - 3))}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.6

Subtract $3$ from $0$ .

$\frac{27 - 1 - 9 + 3 (- - 3)}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.7

Multiply $3 (- - 3)$ .

Tap for more steps...

Step 3.5.7.1

Multiply $- 1$ by $- 3$ .

$\frac{27 - 1 - 9 + 3 \cdot 3}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.7.2

Multiply $3$ by $3$ .

$\frac{27 - 1 - 9 + 9}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.8

Subtract $1$ from $27$ .

$\frac{26 - 9 + 9}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.9

Subtract $9$ from $26$ .

$\frac{17 + 9}{3 \cdot 3 + 3 (- (1))}$

Step 3.5.10

Add $17$ and $9$ .

$\frac{26}{3 \cdot 3 + 3 (- (1))}$

Step 3.6

Simplify the denominator.

Tap for more steps...

Step 3.6.1

Multiply $3$ by $3$ .

$\frac{26}{9 + 3 (- (1))}$

Step 3.6.2

Multiply $3 (- (1))$ .

Tap for more steps...

Step 3.6.2.1

Multiply $- 1$ by $1$ .

$\frac{26}{9 + 3 \cdot - 1}$

Step 3.6.2.2

Multiply $3$ by $- 1$ .

$\frac{26}{9 - 3}$

Step 3.6.3

Subtract $3$ from $9$ .

$\frac{26}{6}$

Step 3.7

Cancel the common factor of $26$ and $6$ .

Tap for more steps...

Step 3.7.1

Factor $2$ out of $26$ .

$\frac{2 (13)}{6}$

Step 3.7.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 13}{2 \cdot 3}$

Step 3.7.2.2

Cancel the common factor.

$\frac{2 \cdot 13}{2 \cdot 3}$

Step 3.7.2.3

Rewrite the expression.

$\frac{13}{3}$