Precalculus Examples

g (x) = 2 x^{3} - 3

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

- 3

$- 3$ ,

3

$3$

Step 1

Write

g (x) = 2 x^{3} - 3

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

y = 2 x^{3} - 3

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

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

$y$ values of the two points divided by the change in

x

$x$ values of the two points.

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

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

Step 2.2

Substitute the equation

y = 2 x^{3} - 3

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

f (3)

$f (3)$ and

f (- 3)

$f (- 3)$ , replacing

x

$x$ in the function with the corresponding

x

$x$ value.

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

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

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

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

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

$3$ to the power of

3

$3$ .

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

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

Step 3.1.2

Multiply

2

$2$ by

27

$27$ .

\frac{54 - 3 - (2 {(- 3)}^{3} - 3)}{3 - (- 3)}

$\frac{54 - 3 - (2 {(- 3)}^{3} - 3)}{3 - (- 3)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Raise

- 3

$- 3$ to the power of

3

$3$ .

\frac{54 - 3 - (2 \cdot - 27 - 3)}{3 - (- 3)}

$\frac{54 - 3 - (2 \cdot - 27 - 3)}{3 - (- 3)}$

Step 3.1.3.2

Multiply

2

$2$ by

- 27

$- 27$ .

\frac{54 - 3 - (- 54 - 3)}{3 - (- 3)}

$\frac{54 - 3 - (- 54 - 3)}{3 - (- 3)}$

\frac{54 - 3 - (- 54 - 3)}{3 - (- 3)}

$\frac{54 - 3 - (- 54 - 3)}{3 - (- 3)}$

Step 3.1.4

Subtract

3

$3$ from

- 54

$- 54$ .

\frac{54 - 3 - - 57}{3 - (- 3)}

$\frac{54 - 3 - - 57}{3 - (- 3)}$

Step 3.1.5

Multiply

- 1

$- 1$ by

- 57

$- 57$ .

\frac{54 - 3 + 57}{3 - (- 3)}

$\frac{54 - 3 + 57}{3 - (- 3)}$

Step 3.1.6

Subtract

3

$3$ from

54

$54$ .

\frac{51 + 57}{3 - (- 3)}

$\frac{51 + 57}{3 - (- 3)}$

Step 3.1.7

Add

51

$51$ and

57

$57$ .

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

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

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

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

\frac{108}{3 + 3}

$\frac{108}{3 + 3}$

Step 3.2.2

Add

3

$3$ and

3

$3$ .

\frac{108}{6}

$\frac{108}{6}$

\frac{108}{6}

$\frac{108}{6}$

Step 3.3

Divide

108

$108$ by

6

$6$ .

18

$18$

18

$18$