Precalculus Examples

$g (x) = 4 x^{3} - 2$ on $- 3$ , $3$

Step 1

Write $g (x) = 4 x^{3} - 2$ as an equation.

$y = 4 x^{3} - 2$

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.

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

Step 2.2

Substitute the equation $y = 4 x^{3} - 2$ for $f (3)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(4 {(3)}^{3} - 2) - (4 {(- 3)}^{3} - 2)}{(3) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{4 \cdot 27 - 2 - (4 {(- 3)}^{3} - 2)}{3 - (- 3)}$

Step 3.1.2

Multiply $4$ by $27$ .

$\frac{108 - 2 - (4 {(- 3)}^{3} - 2)}{3 - (- 3)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

$\frac{108 - 2 - (4 \cdot - 27 - 2)}{3 - (- 3)}$

Step 3.1.3.2

Multiply $4$ by $- 27$ .

$\frac{108 - 2 - (- 108 - 2)}{3 - (- 3)}$

Step 3.1.4

Subtract $2$ from $- 108$ .

$\frac{108 - 2 - - 110}{3 - (- 3)}$

Step 3.1.5

Multiply $- 1$ by $- 110$ .

$\frac{108 - 2 + 110}{3 - (- 3)}$

Step 3.1.6

Subtract $2$ from $108$ .

$\frac{106 + 110}{3 - (- 3)}$

Step 3.1.7

Add $106$ and $110$ .

$\frac{216}{3 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{216}{3 + 3}$

Step 3.2.2

Add $3$ and $3$ .

$\frac{216}{6}$

Step 3.3

Divide $216$ by $6$ .

$36$