Find g(f(x)) f(x)=5x+3 and g(x)=6x^2-5x

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

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 $g$ .

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

Step 3

Simplify each term.

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

$g (5 x + 3) = 6 ((5 x + 3) (5 x + 3)) - 5 (5 x + 3)$

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

Apply the distributive property.

$g (5 x + 3) = 6 (5 x (5 x + 3) + 3 (5 x + 3)) - 5 (5 x + 3)$

Apply the distributive property.

$g (5 x + 3) = 6 (5 x (5 x) + 5 x \cdot 3 + 3 (5 x + 3)) - 5 (5 x + 3)$

Apply the distributive property.

$g (5 x + 3) = 6 (5 x (5 x) + 5 x \cdot 3 + 3 (5 x) + 3 \cdot 3) - 5 (5 x + 3)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (5 x + 3) = 6 (5 \cdot (5 x \cdot x) + 5 x \cdot 3 + 3 (5 x) + 3 \cdot 3) - 5 (5 x + 3)$

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

Move $x$ .

$g (5 x + 3) = 6 (5 \cdot (5 (x \cdot x)) + 5 x \cdot 3 + 3 (5 x) + 3 \cdot 3) - 5 (5 x + 3)$

Multiply $x$ by $x$ .

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

Multiply $5$ by $5$ .

$g (5 x + 3) = 6 (25 x^{2} + 5 x \cdot 3 + 3 (5 x) + 3 \cdot 3) - 5 (5 x + 3)$

Multiply $3$ by $5$ .

$g (5 x + 3) = 6 (25 x^{2} + 15 x + 3 (5 x) + 3 \cdot 3) - 5 (5 x + 3)$

Multiply $5$ by $3$ .

$g (5 x + 3) = 6 (25 x^{2} + 15 x + 15 x + 3 \cdot 3) - 5 (5 x + 3)$

Multiply $3$ by $3$ .

$g (5 x + 3) = 6 (25 x^{2} + 15 x + 15 x + 9) - 5 (5 x + 3)$

Add $15 x$ and $15 x$ .

$g (5 x + 3) = 6 (25 x^{2} + 30 x + 9) - 5 (5 x + 3)$

Apply the distributive property.

$g (5 x + 3) = 6 (25 x^{2}) + 6 (30 x) + 6 \cdot 9 - 5 (5 x + 3)$

Simplify.

Multiply $25$ by $6$ .

$g (5 x + 3) = 150 x^{2} + 6 (30 x) + 6 \cdot 9 - 5 (5 x + 3)$

Multiply $30$ by $6$ .

$g (5 x + 3) = 150 x^{2} + 180 x + 6 \cdot 9 - 5 (5 x + 3)$

Multiply $6$ by $9$ .

$g (5 x + 3) = 150 x^{2} + 180 x + 54 - 5 (5 x + 3)$

Apply the distributive property.

$g (5 x + 3) = 150 x^{2} + 180 x + 54 - 5 (5 x) - 5 \cdot 3$

Multiply $5$ by $- 5$ .

$g (5 x + 3) = 150 x^{2} + 180 x + 54 - 25 x - 5 \cdot 3$

Multiply $- 5$ by $3$ .

$g (5 x + 3) = 150 x^{2} + 180 x + 54 - 25 x - 15$

Step 4

Simplify by adding terms.

Subtract $25 x$ from $180 x$ .

$g (5 x + 3) = 150 x^{2} + 155 x + 54 - 15$

Subtract $15$ from $54$ .

$g (5 x + 3) = 150 x^{2} + 155 x + 39$

Find g(f(x)) f(x)=5x+3 and g(x)=6x^2-5x

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