Precalculus Examples

$f (x) = e^{x - 3} + 2$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$f (x + h) = e^{(x + h) - 3} + 2$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = e^{x + h - 3} + 2$

Step 2.1.2.2

The final answer is $e^{x + h - 3} + 2$ .

$e^{x + h - 3} + 2$

Step 2.2

Find the components of the definition.

$f (x + h) = e^{x + h - 3} + 2$

$f (x) = e^{x - 3} + 2$

$f (x + h) = e^{x + h - 3} + 2$

$f (x) = e^{x - 3} + 2$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{e^{x + h - 3} + 2 - (e^{x - 3} + 2)}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{e^{x + h - 3} + 2 - e^{x - 3} - 1 \cdot 2}{h}$

Step 4.2

Multiply $- 1$ by $2$ .

$\frac{e^{x + h - 3} + 2 - e^{x - 3} - 2}{h}$

Step 4.3

Subtract $2$ from $2$ .

$\frac{e^{x + h - 3} + 0 - e^{x - 3}}{h}$

Step 4.4

Add $e^{x + h - 3}$ and $0$ .

$\frac{e^{x + h - 3} - e^{x - 3}}{h}$

Step 5