Precalculus Examples

$k (x) = 10^{x}$ , given $- 3 \leq x \leq 1$

Step 1

Write $k (x) = 10^{x}$ as an equation.

$y = 10^{x}$

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.

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

Step 2.2

Substitute the equation $y = 10^{x}$ for $f (1)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(10^{1}) - (10^{- 3})}{(1) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Evaluate the exponent.

$\frac{10 - 10^{- 3}}{1 - (- 3)}$

Step 3.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10 - \frac{1}{10^{3}}}{1 - (- 3)}$

Step 3.1.3

Raise $10$ to the power of $3$ .

$\frac{10 - \frac{1}{1000}}{1 - (- 3)}$

Step 3.1.4

To write $10$ as a fraction with a common denominator, multiply by $\frac{1000}{1000}$ .

$\frac{10 \cdot \frac{1000}{1000} - \frac{1}{1000}}{1 - (- 3)}$

Step 3.1.5

Combine $10$ and $\frac{1000}{1000}$ .

$\frac{\frac{10 \cdot 1000}{1000} - \frac{1}{1000}}{1 - (- 3)}$

Step 3.1.6

Combine the numerators over the common denominator.

$\frac{\frac{10 \cdot 1000 - 1}{1000}}{1 - (- 3)}$

Step 3.1.7

Simplify the numerator.

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Step 3.1.7.1

Multiply $10$ by $1000$ .

$\frac{\frac{10000 - 1}{1000}}{1 - (- 3)}$

Step 3.1.7.2

Subtract $1$ from $10000$ .

$\frac{\frac{9999}{1000}}{1 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{\frac{9999}{1000}}{1 + 3}$

Step 3.2.2

Add $1$ and $3$ .

$\frac{\frac{9999}{1000}}{4}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{9999}{1000} \cdot \frac{1}{4}$

Step 3.4

Multiply $\frac{9999}{1000} \cdot \frac{1}{4}$ .

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Step 3.4.1

Multiply $\frac{9999}{1000}$ by $\frac{1}{4}$ .

$\frac{9999}{1000 \cdot 4}$

Step 3.4.2

Multiply $1000$ by $4$ .

$\frac{9999}{4000}$