Find g(f(x)) g(x)=4-3/2x f(x)=1/2x+3/2

$g (x) = 4 - \frac{3}{2} x$ $f (x) = \frac{1}{2} x + \frac{3}{2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{1}{2} x + \frac{3}{2})$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

Combine $\frac{1}{2}$ and $x$ .

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

Apply the distributive property.

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $3$ by $3$ .

$g (\frac{1}{2} x + \frac{3}{2}) = 4 - \frac{x \cdot 3}{4} - \frac{9}{2 \cdot 2}$

Multiply $2$ by $2$ .

$g (\frac{1}{2} x + \frac{3}{2}) = 4 - \frac{x \cdot 3}{4} - \frac{9}{4}$

Move $3$ to the left of $x$ .

$g (\frac{1}{2} x + \frac{3}{2}) = 4 - \frac{3 x}{4} - \frac{9}{4}$

Step 4

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$g (\frac{1}{2} x + \frac{3}{2}) = - \frac{3 x}{4} + 4 \cdot \frac{4}{4} - \frac{9}{4}$

Step 5

Combine $4$ and $\frac{4}{4}$ .

$g (\frac{1}{2} x + \frac{3}{2}) = - \frac{3 x}{4} + \frac{4 \cdot 4}{4} - \frac{9}{4}$

Step 6

Combine the numerators over the common denominator.

$g (\frac{1}{2} x + \frac{3}{2}) = - \frac{3 x}{4} + \frac{4 \cdot 4 - 9}{4}$

Step 7

Simplify the numerator.

Multiply $4$ by $4$ .

$g (\frac{1}{2} x + \frac{3}{2}) = - \frac{3 x}{4} + \frac{16 - 9}{4}$

Subtract $9$ from $16$ .

$g (\frac{1}{2} x + \frac{3}{2}) = - \frac{3 x}{4} + \frac{7}{4}$