Precalculus Examples

$f (x) = \frac{2}{3} x + 1$ , $f^{- 1} (x) = \frac{3 x - 3}{2}$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$f (x) = \frac{2 x}{3} + 1$

Step 2

Set up the long division problem to evaluate the function at $\frac{3 x - 3}{2}$ .

$\frac{\frac{2 x}{3} + 1}{x - (\frac{3 x - 3}{2})}$

Step 3

Divide using synthetic division.

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Step 3.1

Divide each term in the denominator by $\frac{- 1}{2}$ to make the coefficient of linear factor variable $1$ .

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

Step 3.2

Place the numbers representing the divisor and the dividend into a division-like configuration.

$3$	$\frac{2}{3}$	$1$

Step 3.3

The first number in the dividend $(\frac{2}{3})$ is put into the first position of the result area (below the horizontal line).

$3$	$\frac{2}{3}$	$1$

	$\frac{2}{3}$

Step 3.4

Multiply the newest entry in the result $(\frac{2}{3})$ by the divisor $(3)$ and place the result of $(2)$ under the next term in the dividend $(1)$ .

$3$	$\frac{2}{3}$	$1$
		$2$
	$\frac{2}{3}$

Step 3.5

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$\frac{2}{3}$	$1$
		$2$
	$\frac{2}{3}$	$3$

Step 3.6

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Move the negative in front of the fraction.

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

Step 3.7.3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.7.3.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.7.3.2

Move the negative in front of the fraction.

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

Step 3.7.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.7.5

Multiply $- (1 \cdot 2)$ .

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Step 3.7.5.1

Multiply $2$ by $1$ .

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

Step 3.7.5.2

Multiply $- 1$ by $2$ .

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

Step 3.7.6

Multiply $- 2$ by $\frac{2}{3}$ .

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

Step 3.7.7

Multiply $- 2 (\frac{2}{3})$ .

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Step 3.7.7.1

Combine $- 2$ and $\frac{2}{3}$ .

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

Step 3.7.7.2

Multiply $- 2$ by $2$ .

$\frac{- 4}{3} + \frac{1}{\frac{- 1}{2}} \cdot \frac{3}{x - 3}$

Step 3.7.8

Move the negative in front of the fraction.

$- \frac{4}{3} + \frac{1}{\frac{- 1}{2}} \cdot \frac{3}{x - 3}$

Step 3.7.9

Move the negative in front of the fraction.

$- \frac{4}{3} + \frac{1}{- \frac{1}{2}} \cdot \frac{3}{x - 3}$

Step 3.7.10

Cancel the common factor of $1$ and $- 1$ .

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Step 3.7.10.1

Rewrite $1$ as $- 1 (- 1)$ .

$- \frac{4}{3} + \frac{- 1 (- 1)}{- \frac{1}{2}} \cdot \frac{3}{x - 3}$

Step 3.7.10.2

Move the negative in front of the fraction.

$- \frac{4}{3} - \frac{1}{\frac{1}{2}} \cdot \frac{3}{x - 3}$

Step 3.7.11

Multiply the numerator by the reciprocal of the denominator.

$- \frac{4}{3} - (1 \cdot 2) \cdot \frac{3}{x - 3}$

Step 3.7.12

Multiply $- (1 \cdot 2)$ .

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Step 3.7.12.1

Multiply $2$ by $1$ .

$- \frac{4}{3} - 1 \cdot 2 \cdot \frac{3}{x - 3}$

Step 3.7.12.2

Multiply $- 1$ by $2$ .

$- \frac{4}{3} - 2 \cdot \frac{3}{x - 3}$

Step 3.7.13

Multiply $- 2$ by $\frac{3}{x - 3}$ .

$- \frac{4}{3} - 2 \frac{3}{x - 3}$

Step 3.7.14

Multiply $- 2 \frac{3}{x - 3}$ .

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Step 3.7.14.1

Combine $- 2$ and $\frac{3}{x - 3}$ .

$- \frac{4}{3} + \frac{- 2 \cdot 3}{x - 3}$

Step 3.7.14.2

Multiply $- 2$ by $3$ .

$- \frac{4}{3} + \frac{- 6}{x - 3}$

Step 3.7.15

Move the negative in front of the fraction.

$- \frac{4}{3} - \frac{6}{x - 3}$

Step 4

The remainder of the synthetic division is the result based on the remainder theorem.

$3$

Step 5