Precalculus Examples

$f (6) = \frac{1000}{6^{2} + 4}$

Step 1

Write $f (6) - 25 = 0$ as a function.

$f (x) = f (6) - 25$

Step 2

Subtract $\frac{1000}{6^{2} + 4}$ from both sides of the equation.

$f (6) - \frac{1000}{6^{2} + 4} = 0$

Step 3

Simplify each term.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Raise $6$ to the power of $2$ .

$f (6) - \frac{1000}{36 + 4} = 0$

Step 3.1.2

Add $36$ and $4$ .

$f (6) - \frac{1000}{40} = 0$

Step 3.2

Divide $1000$ by $40$ .

$f (6) - 1 \cdot 25 = 0$

Step 3.3

Multiply $- 1$ by $25$ .

$f (6) - 25 = 0$

Step 4

Consider the difference quotient formula.

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

Step 5

Find the components of the definition.

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Step 5.1

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

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Step 5.1.1

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

$f (x + h) = f (6) - 25$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Move $6$ to the left of $f$ .

$f (x + h) = 6 f - 25$

Step 5.1.2.2

The final answer is $6 f - 25$ .

$6 f - 25$

Step 5.2

Find the components of the definition.

$f (x + h) = 6 f - 25$

$f (x) = 6 f - 25$

$f (x + h) = 6 f - 25$

$f (x) = 6 f - 25$

Step 6

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{6 f - 25 - (6 f - 25)}{h}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$\frac{6 f - 25 - (6 f) - - 25}{h}$

Step 7.1.2

Multiply $6$ by $- 1$ .

$\frac{6 f - 25 - 6 f - - 25}{h}$

Step 7.1.3

Multiply $- 1$ by $- 25$ .

$\frac{6 f - 25 - 6 f + 25}{h}$

Step 7.1.4

Subtract $6 f$ from $6 f$ .

$\frac{0 - 25 + 25}{h}$

Step 7.1.5

Subtract $25$ from $0$ .

$\frac{- 25 + 25}{h}$

Step 7.1.6

Add $- 25$ and $25$ .

$\frac{0}{h}$

Step 7.2

Divide $0$ by $h$ .

$0$

Step 8