Find f(h(x)) f(x)=6-1/2x h(x)=4(x-3)^2

$f (x) = 6 - \frac{1}{2} x$ $h (x) = 4 {(x - 3)}^{2}$

Step 1

Set up the composite result function.

$f (h (x))$

Step 2

Evaluate $f (4 {(x - 3)}^{2})$ by substituting in the value of $h$ into $f$ .

$f (4 {(x - 3)}^{2}) = 6 - \frac{1}{2} \cdot (4 {(x - 3)}^{2})$

Step 3

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (4 {(x - 3)}^{2}) = 6 + \frac{- 1}{2} \cdot (4 {(x - 3)}^{2})$

Factor $2$ out of $4 {(x - 3)}^{2}$ .

$f (4 {(x - 3)}^{2}) = 6 + \frac{- 1}{2} \cdot (2 (2 {(x - 3)}^{2}))$

Cancel the common factor.

$f (4 {(x - 3)}^{2}) = 6 + \frac{- 1}{2} \cdot (2 (2 {(x - 3)}^{2}))$

Rewrite the expression.

$f (4 {(x - 3)}^{2}) = 6 - 1 \cdot (2 {(x - 3)}^{2})$

Multiply $2$ by $- 1$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot {(x - 3)}^{2}$

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot ((x - 3) (x - 3))$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x (x - 3) - 3 (x - 3))$

Apply the distributive property.

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Apply the distributive property.

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

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

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Multiply $- 3$ by $- 3$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x^{2} - 3 x - 3 x + 9)$

Subtract $3 x$ from $- 3 x$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 \cdot (x^{2} - 6 x + 9)$

Apply the distributive property.

$f (4 {(x - 3)}^{2}) = 6 - 2 x^{2} - 2 (- 6 x) - 2 \cdot 9$

Simplify.

Multiply $- 6$ by $- 2$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 x^{2} + 12 x - 2 \cdot 9$

Multiply $- 2$ by $9$ .

$f (4 {(x - 3)}^{2}) = 6 - 2 x^{2} + 12 x - 18$

Step 4

Subtract $18$ from $6$ .

$f (4 {(x - 3)}^{2}) = - 2 x^{2} + 12 x - 12$

Find f(h(x)) f(x)=6-1/2x h(x)=4(x-3)^2

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