Find g(f(x)) f(x)=x-4 and g(x)=5x^2-2

$f (x) = x - 4$ and $g (x) = 5 x^{2} - 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x - 4)$ by substituting in the value of $f$ into $g$ .

$g (x - 4) = 5 {(x - 4)}^{2} - 2$

Step 3

Simplify each term.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$g (x - 4) = 5 ((x - 4) (x - 4)) - 2$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$g (x - 4) = 5 (x (x - 4) - 4 (x - 4)) - 2$

Apply the distributive property.

$g (x - 4) = 5 (x \cdot x + x \cdot - 4 - 4 (x - 4)) - 2$

Apply the distributive property.

$g (x - 4) = 5 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$g (x - 4) = 5 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) - 2$

Move $- 4$ to the left of $x$ .

$g (x - 4) = 5 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) - 2$

Multiply $- 4$ by $- 4$ .

$g (x - 4) = 5 (x^{2} - 4 x - 4 x + 16) - 2$

Subtract $4 x$ from $- 4 x$ .

$g (x - 4) = 5 (x^{2} - 8 x + 16) - 2$

Apply the distributive property.

$g (x - 4) = 5 x^{2} + 5 (- 8 x) + 5 \cdot 16 - 2$

Simplify.

Multiply $- 8$ by $5$ .

$g (x - 4) = 5 x^{2} - 40 x + 5 \cdot 16 - 2$

Multiply $5$ by $16$ .

$g (x - 4) = 5 x^{2} - 40 x + 80 - 2$

Step 4

Subtract $2$ from $80$ .

$g (x - 4) = 5 x^{2} - 40 x + 78$