Precalculus Examples

$y = - \sqrt{x - 1} + 5$

Step 1

Write $y = - \sqrt{x - 1} + 5$ as a function.

$f (x) = - \sqrt{x - 1} + 5$

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.

$f (x + h) = - \sqrt{(x + h) - 1} + 5$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Remove parentheses.

$f (x + h) = - \sqrt{x + h - 1} + 5$

Step 3.1.2.2

The final answer is $- \sqrt{x + h - 1} + 5$ .

$- \sqrt{x + h - 1} + 5$

Step 3.2

Find the components of the definition.

$f (x + h) = - \sqrt{x + h - 1} + 5$

$f (x) = - \sqrt{x - 1} + 5$

$f (x + h) = - \sqrt{x + h - 1} + 5$

$f (x) = - \sqrt{x - 1} + 5$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \sqrt{x + h - 1} + 5 - (- \sqrt{x - 1} + 5)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{- \sqrt{x + h - 1} + 5 - - \sqrt{x - 1} - 1 \cdot 5}{h}$

Step 5.1.2

Multiply $- - \sqrt{x - 1}$ .

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Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt{x + h - 1} + 5 + 1 \sqrt{x - 1} - 1 \cdot 5}{h}$

Step 5.1.2.2

Multiply $\sqrt{x - 1}$ by $1$ .

$\frac{- \sqrt{x + h - 1} + 5 + \sqrt{x - 1} - 1 \cdot 5}{h}$

Step 5.1.3

Multiply $- 1$ by $5$ .

$\frac{- \sqrt{x + h - 1} + 5 + \sqrt{x - 1} - 5}{h}$

Step 5.1.4

Subtract $5$ from $5$ .

$\frac{- \sqrt{x + h - 1} + 0 + \sqrt{x - 1}}{h}$

Step 5.1.5

Add $- \sqrt{x + h - 1}$ and $0$ .

$\frac{- \sqrt{x + h - 1} + \sqrt{x - 1}}{h}$

Step 5.2

Simplify with factoring out.

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Step 5.2.1

Factor $- 1$ out of $\sqrt{x - 1}$ .

$\frac{- \sqrt{x + h - 1} - 1 (- \sqrt{x - 1})}{h}$

Step 5.2.2

Factor $- 1$ out of $- \sqrt{x + h - 1} - 1 (- \sqrt{x - 1})$ .

$\frac{- (\sqrt{x + h - 1} - \sqrt{x - 1})}{h}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Rewrite $- (\sqrt{x + h - 1} - \sqrt{x - 1})$ as $- 1 (\sqrt{x + h - 1} - \sqrt{x - 1})$ .

$\frac{- 1 (\sqrt{x + h - 1} - \sqrt{x - 1})}{h}$

Step 5.2.3.2

Move the negative in front of the fraction.

$- \frac{\sqrt{x + h - 1} - \sqrt{x - 1}}{h}$

Step 6