Find g(f(x)) f(x)=5x-1 , g(x)=x^2-2

$f (x) = 5 x - 1$ , $g (x) = x^{2} - 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (5 x - 1)$ by substituting in the value of $f$ into $g$ .

$g (5 x - 1) = {(5 x - 1)}^{2} - 2$

Step 3

Simplify each term.

Rewrite ${(5 x - 1)}^{2}$ as $(5 x - 1) (5 x - 1)$ .

$g (5 x - 1) = (5 x - 1) (5 x - 1) - 2$

Expand $(5 x - 1) (5 x - 1)$ using the FOIL Method.

Apply the distributive property.

$g (5 x - 1) = 5 x (5 x - 1) - 1 (5 x - 1) - 2$

Apply the distributive property.

$g (5 x - 1) = 5 x (5 x) + 5 x \cdot - 1 - 1 (5 x - 1) - 2$

Apply the distributive property.

$g (5 x - 1) = 5 x (5 x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1 - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (5 x - 1) = 5 \cdot (5 x \cdot x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1 - 2$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$g (5 x - 1) = 5 \cdot (5 (x \cdot x)) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1 - 2$

Multiply $x$ by $x$ .

$g (5 x - 1) = 5 \cdot (5 x^{2}) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1 - 2$

Multiply $5$ by $5$ .

$g (5 x - 1) = 25 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1 - 2$

Multiply $- 1$ by $5$ .

$g (5 x - 1) = 25 x^{2} - 5 x - 1 (5 x) - 1 \cdot - 1 - 2$

Multiply $5$ by $- 1$ .

$g (5 x - 1) = 25 x^{2} - 5 x - 5 x - 1 \cdot - 1 - 2$

Multiply $- 1$ by $- 1$ .

$g (5 x - 1) = 25 x^{2} - 5 x - 5 x + 1 - 2$

Subtract $5 x$ from $- 5 x$ .

$g (5 x - 1) = 25 x^{2} - 10 x + 1 - 2$

Step 4

Subtract $2$ from $1$ .

$g (5 x - 1) = 25 x^{2} - 10 x - 1$