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

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (5 x - 3)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

Rewrite ${(5 x - 3)}^{2}$ as $(5 x - 3) (5 x - 3)$ .

$g (5 x - 3) = 2 ((5 x - 3) (5 x - 3)) - 5$

Expand $(5 x - 3) (5 x - 3)$ using the FOIL Method.

Apply the distributive property.

$g (5 x - 3) = 2 (5 x (5 x - 3) - 3 (5 x - 3)) - 5$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $5$ by $5$ .

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

Multiply $- 3$ by $5$ .

$g (5 x - 3) = 2 (25 x^{2} - 15 x - 3 (5 x) - 3 \cdot - 3) - 5$

Multiply $5$ by $- 3$ .

$g (5 x - 3) = 2 (25 x^{2} - 15 x - 15 x - 3 \cdot - 3) - 5$

Multiply $- 3$ by $- 3$ .

$g (5 x - 3) = 2 (25 x^{2} - 15 x - 15 x + 9) - 5$

Subtract $15 x$ from $- 15 x$ .

$g (5 x - 3) = 2 (25 x^{2} - 30 x + 9) - 5$

Apply the distributive property.

$g (5 x - 3) = 2 (25 x^{2}) + 2 (- 30 x) + 2 \cdot 9 - 5$

Simplify.

Multiply $25$ by $2$ .

$g (5 x - 3) = 50 x^{2} + 2 (- 30 x) + 2 \cdot 9 - 5$

Multiply $- 30$ by $2$ .

$g (5 x - 3) = 50 x^{2} - 60 x + 2 \cdot 9 - 5$

Multiply $2$ by $9$ .

$g (5 x - 3) = 50 x^{2} - 60 x + 18 - 5$

Step 4

Subtract $5$ from $18$ .

$g (5 x - 3) = 50 x^{2} - 60 x + 13$

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

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