Find g(f(x)) f(x)=5x-1 and g(x)=4x^2-3

$f (x) = 5 x - 1$ and $g (x) = 4 x^{2} - 3$

Set up the composite result function.

$g (f (x))$

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

$g (5 x - 1) = 4 {(5 x - 1)}^{2} - 3$

Simplify each term.

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

$g (5 x - 1) = 4 ((5 x - 1) (5 x - 1)) - 3$

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

Apply the distributive property.

$g (5 x - 1) = 4 (5 x (5 x - 1) - 1 (5 x - 1)) - 3$

Apply the distributive property.

$g (5 x - 1) = 4 (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x - 1)) - 3$

Apply the distributive property.

$g (5 x - 1) = 4 (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (5 x - 1) = 4 (5 \cdot (5 x \cdot x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) - 3$

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

Move $x$ .

$g (5 x - 1) = 4 (5 \cdot (5 (x \cdot x)) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) - 3$

Multiply $x$ by $x$ .

$g (5 x - 1) = 4 (5 \cdot (5 x^{2}) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) - 3$

Multiply $5$ by $5$ .

$g (5 x - 1) = 4 (25 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) - 3$

Multiply $- 1$ by $5$ .

$g (5 x - 1) = 4 (25 x^{2} - 5 x - 1 (5 x) - 1 \cdot - 1) - 3$

Multiply $5$ by $- 1$ .

$g (5 x - 1) = 4 (25 x^{2} - 5 x - 5 x - 1 \cdot - 1) - 3$

Multiply $- 1$ by $- 1$ .

$g (5 x - 1) = 4 (25 x^{2} - 5 x - 5 x + 1) - 3$

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

$g (5 x - 1) = 4 (25 x^{2} - 10 x + 1) - 3$

Apply the distributive property.

$g (5 x - 1) = 4 (25 x^{2}) + 4 (- 10 x) + 4 \cdot 1 - 3$

Simplify.

Multiply $25$ by $4$ .

$g (5 x - 1) = 100 x^{2} + 4 (- 10 x) + 4 \cdot 1 - 3$

Multiply $- 10$ by $4$ .

$g (5 x - 1) = 100 x^{2} - 40 x + 4 \cdot 1 - 3$

Multiply $4$ by $1$ .

$g (5 x - 1) = 100 x^{2} - 40 x + 4 - 3$

Subtract $3$ from $4$ .

$g (5 x - 1) = 100 x^{2} - 40 x + 1$