Precalculus Examples

$y = 128 {(\frac{1}{2})}^{x}$

Step 1

Write $y = 128 {(\frac{1}{2})}^{x}$ as a function.

$f (x) = 128 {(\frac{1}{2})}^{x}$

Step 2

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = x + h$ .

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Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 128 {(\frac{1}{2})}^{x + h}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the expression.

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Step 3.1.2.1.1

Apply the product rule to $\frac{1}{2}$ .

$f (x + h) = 128 (\frac{1^{x + h}}{2^{x + h}})$

Step 3.1.2.1.2

One to any power is one.

$f (x + h) = 128 (\frac{1}{2^{x + h}})$

Step 3.1.2.2

Combine $128$ and $\frac{1}{2^{x + h}}$ .

$f (x + h) = \frac{128}{2^{x + h}}$

Step 3.1.2.3

The final answer is $\frac{128}{2^{x + h}}$ .

$\frac{128}{2^{x + h}}$

Step 3.2

Find the components of the definition.

$f (x + h) = \frac{128}{2^{x + h}}$

$f (x) = \frac{128}{2^{x}}$

$f (x + h) = \frac{128}{2^{x + h}}$

$f (x) = \frac{128}{2^{x}}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{128}{2^{x + h}} - (\frac{128}{2^{x}})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

To write $\frac{128}{2^{x + h}}$ as a fraction with a common denominator, multiply by $\frac{2^{x}}{2^{x}}$ .

$\frac{\frac{128}{2^{x + h}} \cdot \frac{2^{x}}{2^{x}} - \frac{128}{2^{x}}}{h}$

Step 5.1.2

To write $- \frac{128}{2^{x}}$ as a fraction with a common denominator, multiply by $\frac{2^{h + x}}{2^{h + x}}$ .

$\frac{\frac{128}{2^{x + h}} \cdot \frac{2^{x}}{2^{x}} - \frac{128}{2^{x}} \cdot \frac{2^{h + x}}{2^{h + x}}}{h}$

Step 5.1.3

Write each expression with a common denominator of $2^{x + h} \cdot 2^{x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1.3.1

Multiply $\frac{128}{2^{x + h}}$ by $\frac{2^{x}}{2^{x}}$ .

$\frac{\frac{128 \cdot 2^{x}}{2^{x + h} \cdot 2^{x}} - \frac{128}{2^{x}} \cdot \frac{2^{h + x}}{2^{h + x}}}{h}$

Step 5.1.3.2

Multiply $2^{x + h}$ by $2^{x}$ by adding the exponents.

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Step 5.1.3.2.1

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

$\frac{\frac{128 \cdot 2^{x}}{2^{x + h + x}} - \frac{128}{2^{x}} \cdot \frac{2^{h + x}}{2^{h + x}}}{h}$

Step 5.1.3.2.2

Add $x$ and $x$ .

$\frac{\frac{128 \cdot 2^{x}}{2^{2 x + h}} - \frac{128}{2^{x}} \cdot \frac{2^{h + x}}{2^{h + x}}}{h}$

Step 5.1.3.3

Multiply $\frac{128}{2^{x}}$ by $\frac{2^{h + x}}{2^{h + x}}$ .

$\frac{\frac{128 \cdot 2^{x}}{2^{2 x + h}} - \frac{128 \cdot 2^{h + x}}{2^{x} \cdot 2^{h + x}}}{h}$

Step 5.1.3.4

Multiply $2^{x}$ by $2^{h + x}$ by adding the exponents.

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Step 5.1.3.4.1

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

$\frac{\frac{128 \cdot 2^{x}}{2^{2 x + h}} - \frac{128 \cdot 2^{h + x}}{2^{x + h + x}}}{h}$

Step 5.1.3.4.2

Add $x$ and $x$ .

$\frac{\frac{128 \cdot 2^{x}}{2^{2 x + h}} - \frac{128 \cdot 2^{h + x}}{2^{2 x + h}}}{h}$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{\frac{128 \cdot 2^{x} - 128 \cdot 2^{h + x}}{2^{2 x + h}}}{h}$

Step 5.1.5

Factor $128$ out of $128 \cdot 2^{x} - 128 \cdot 2^{h + x}$ .

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Step 5.1.5.1

Factor $128$ out of $128 \cdot 2^{x}$ .

$\frac{\frac{128 (2^{x}) - 128 \cdot 2^{h + x}}{2^{2 x + h}}}{h}$

Step 5.1.5.2

Factor $128$ out of $- 128 \cdot 2^{h + x}$ .

$\frac{\frac{128 (2^{x}) + 128 (- 2^{h + x})}{2^{2 x + h}}}{h}$

Step 5.1.5.3

Factor $128$ out of $128 (2^{x}) + 128 (- 2^{h + x})$ .

$\frac{\frac{128 (2^{x} - 2^{h + x})}{2^{2 x + h}}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{128 (2^{x} - 2^{h + x})}{2^{2 x + h}} \cdot \frac{1}{h}$

Step 5.3

Combine.

$\frac{128 (2^{x} - 2^{h + x}) \cdot 1}{2^{2 x + h} h}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $128$ by $1$ .

$\frac{128 (2^{x} - 2^{h + x})}{2^{2 x + h} h}$

Step 5.4.2

Reorder factors in $\frac{128 (2^{x} - 2^{h + x})}{2^{2 x + h} h}$ .

$\frac{128 (2^{x} - 2^{h + x})}{h \cdot 2^{2 x + h}}$

Step 6