Find g(f(x)) f(x)=x-3 and g(x)=5x^2-3

$f (x) = x - 3$ and $g (x) = 5 x^{2} - 3$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x - 3)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

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

$g (x - 3) = 5 ((x - 3) (x - 3)) - 3$

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

Apply the distributive property.

$g (x - 3) = 5 (x (x - 3) - 3 (x - 3)) - 3$

Apply the distributive property.

$g (x - 3) = 5 (x \cdot x + x \cdot - 3 - 3 (x - 3)) - 3$

Apply the distributive property.

$g (x - 3) = 5 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 3$ by $- 3$ .

$g (x - 3) = 5 (x^{2} - 3 x - 3 x + 9) - 3$

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

$g (x - 3) = 5 (x^{2} - 6 x + 9) - 3$

Apply the distributive property.

$g (x - 3) = 5 x^{2} + 5 (- 6 x) + 5 \cdot 9 - 3$

Simplify.

Multiply $- 6$ by $5$ .

$g (x - 3) = 5 x^{2} - 30 x + 5 \cdot 9 - 3$

Multiply $5$ by $9$ .

$g (x - 3) = 5 x^{2} - 30 x + 45 - 3$

Step 4

Subtract $3$ from $45$ .

$g (x - 3) = 5 x^{2} - 30 x + 42$