Algebra Examples

$f (x) = \frac{2}{3} x - 5$ , $g (x) = \frac{3 x - 15}{2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{2}{3} x - 5)$ by substituting in the value of $f$ into $g$ .

$g (\frac{2}{3} x - 5) = \frac{3 (\frac{2}{3} x - 5) - 15}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $3$ out of $3 (\frac{2}{3} x - 5) - 15$ .

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Step 3.1.1

Factor $3$ out of $- 15$ .

$g (\frac{2}{3} x - 5) = \frac{3 (\frac{2}{3} x - 5) + 3 \cdot - 5}{2}$

Step 3.1.2

Factor $3$ out of $3 (\frac{2}{3} x - 5) + 3 \cdot - 5$ .

$g (\frac{2}{3} x - 5) = \frac{3 (\frac{2}{3} x - 5 - 5)}{2}$

Step 3.2

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

$g (\frac{2}{3} x - 5) = \frac{3 (\frac{2 x}{3} - 5 - 5)}{2}$

Step 3.3

Subtract $5$ from $- 5$ .

$g (\frac{2}{3} x - 5) = \frac{3 (\frac{2 x}{3} - 10)}{2}$

Step 3.4

Factor $2$ out of $\frac{2 x}{3} - 10$ .

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Step 3.4.1

Factor $2$ out of $\frac{2 x}{3}$ .

$g (\frac{2}{3} x - 5) = \frac{3 (2 (\frac{x}{3}) - 10)}{2}$

Step 3.4.2

Factor $2$ out of $- 10$ .

$g (\frac{2}{3} x - 5) = \frac{3 (2 (\frac{x}{3}) + 2 \cdot - 5)}{2}$

Step 3.4.3

Factor $2$ out of $2 \frac{x}{3} + 2 \cdot - 5$ .

$g (\frac{2}{3} x - 5) = \frac{3 (2 (\frac{x}{3} - 5))}{2}$

$g (\frac{2}{3} x - 5) = \frac{3 \cdot (2 (\frac{x}{3} - 5))}{2}$

Step 3.5

Multiply $3$ by $2$ .

$g (\frac{2}{3} x - 5) = \frac{6 (\frac{x}{3} - 5)}{2}$

Step 3.6

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$g (\frac{2}{3} x - 5) = \frac{6 (\frac{x}{3} - 5 \cdot \frac{3}{3})}{2}$

Step 3.7

Combine $- 5$ and $\frac{3}{3}$ .

$g (\frac{2}{3} x - 5) = \frac{6 (\frac{x}{3} + \frac{- 5 \cdot 3}{3})}{2}$

Step 3.8

Combine the numerators over the common denominator.

$g (\frac{2}{3} x - 5) = \frac{6 (\frac{x - 5 \cdot 3}{3})}{2}$

Step 3.9

Multiply $- 5$ by $3$ .

$g (\frac{2}{3} x - 5) = \frac{6 (\frac{x - 15}{3})}{2}$

Step 4

Combine $6$ and $\frac{x - 15}{3}$ .

$g (\frac{2}{3} x - 5) = \frac{\frac{6 (x - 15)}{3}}{2}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Reduce the expression $\frac{6 (x - 15)}{3}$ by cancelling the common factors.

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Step 5.1.1

Factor $3$ out of $6 (x - 15)$ .

$g (\frac{2}{3} x - 5) = \frac{\frac{3 (2 (x - 15))}{3}}{2}$

Step 5.1.2

Factor $3$ out of $3$ .

$g (\frac{2}{3} x - 5) = \frac{\frac{3 (2 (x - 15))}{3 (1)}}{2}$

Step 5.1.3

Cancel the common factor.

$g (\frac{2}{3} x - 5) = \frac{\frac{3 (2 (x - 15))}{3 \cdot 1}}{2}$

Step 5.1.4

Rewrite the expression.

$g (\frac{2}{3} x - 5) = \frac{\frac{2 (x - 15)}{1}}{2}$

Step 5.2

Divide $2 (x - 15)$ by $1$ .

$g (\frac{2}{3} x - 5) = \frac{2 (x - 15)}{2}$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$g (\frac{2}{3} x - 5) = \frac{2 (x - 15)}{2}$

Step 6.2

Divide $x - 15$ by $1$ .

$g (\frac{2}{3} x - 5) = x - 15$