Find the Difference Quotient f(x)=6x-9

$f (x) = 6 x - 9$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = 6 (x + h) - 9$

Simplify the result.

Apply the distributive property.

$f (x + h) = 6 x + 6 h - 9$

The final answer is $6 x + 6 h - 9$ .

$6 x + 6 h - 9$

Reorder $6 x$ and $6 h$ .

$6 h + 6 x - 9$

Find the components of the definition.

$f (x + h) = 6 h + 6 x - 9$

$f (x) = 6 x - 9$

$f (x + h) = 6 h + 6 x - 9$

$f (x) = 6 x - 9$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{6 h + 6 x - 9 - (6 x - 9)}{h}$

Step 4

Simplify.

Simplify the numerator.

Factor $3$ out of $6 x - 9$ .

Factor $3$ out of $6 x$ .

$\frac{6 h + 6 x - 9 - (3 (2 x) - 9)}{h}$

Factor $3$ out of $- 9$ .

$\frac{6 h + 6 x - 9 - (3 (2 x) + 3 (- 3))}{h}$

Factor $3$ out of $3 (2 x) + 3 (- 3)$ .

$\frac{6 h + 6 x - 9 - (3 (2 x - 3))}{h}$

$\frac{6 h + 6 x - 9 - 1 \cdot 3 (2 x - 3)}{h}$

Multiply $- 1$ by $3$ .

$\frac{6 h + 6 x - 9 - 3 (2 x - 3)}{h}$

Factor $3$ out of $6 h + 6 x - 9 - 3 (2 x - 3)$ .

Factor $3$ out of $6 h$ .

$\frac{3 (2 h) + 6 x - 9 - 3 (2 x - 3)}{h}$

Factor $3$ out of $6 x$ .

$\frac{3 (2 h) + 3 (2 x) - 9 - 3 (2 x - 3)}{h}$

Factor $3$ out of $- 9$ .

$\frac{3 (2 h) + 3 (2 x) + 3 (- 3) - 3 (2 x - 3)}{h}$

Factor $3$ out of $- 3 (2 x - 3)$ .

$\frac{3 (2 h) + 3 (2 x) + 3 (- 3) + 3 (- (2 x - 3))}{h}$

Factor $3$ out of $3 (2 h) + 3 (2 x)$ .

$\frac{3 (2 h + 2 x) + 3 (- 3) + 3 (- (2 x - 3))}{h}$

Factor $3$ out of $3 (2 h + 2 x) + 3 (- 3)$ .

$\frac{3 (2 h + 2 x - 3) + 3 (- (2 x - 3))}{h}$

Factor $3$ out of $3 (2 h + 2 x - 3) + 3 (- (2 x - 3))$ .

$\frac{3 (2 h + 2 x - 3 - (2 x - 3))}{h}$

Apply the distributive property.

$\frac{3 (2 h + 2 x - 3 - (2 x) - - 3)}{h}$

Multiply $2$ by $- 1$ .

$\frac{3 (2 h + 2 x - 3 - 2 x - - 3)}{h}$

Multiply $- 1$ by $- 3$ .

$\frac{3 (2 h + 2 x - 3 - 2 x + 3)}{h}$

Subtract $2 x$ from $2 x$ .

$\frac{3 (2 h + 0 - 3 + 3)}{h}$

Add $2 h$ and $0$ .

$\frac{3 (2 h - 3 + 3)}{h}$

Add $- 3$ and $3$ .

$\frac{3 (2 h + 0)}{h}$

Add $2 h$ and $0$ .

$\frac{3 \cdot 2 h}{h}$

Multiply $3$ by $2$ .

$\frac{6 h}{h}$

Divide $6$ by $1$ .

$6$

Step 5

Find the Difference Quotient f(x)=6x-9

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