Find f(g(x)) f(x)=2x^2-5x+10 g(x)=-2x-3

$f (x) = 2 x^{2} - 5 x + 10$ $g (x) = - 2 x - 3$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (- 2 x - 3)$ by substituting in the value of $g$ into $f$ .

$f (- 2 x - 3) = 2 {(- 2 x - 3)}^{2} - 5 (- 2 x - 3) + 10$

Step 3

Simplify each term.

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

$f (- 2 x - 3) = 2 ((- 2 x - 3) (- 2 x - 3)) - 5 (- 2 x - 3) + 10$

Expand $(- 2 x - 3) (- 2 x - 3)$ using the FOIL Method.

Apply the distributive property.

$f (- 2 x - 3) = 2 (- 2 x (- 2 x - 3) - 3 (- 2 x - 3)) - 5 (- 2 x - 3) + 10$

Apply the distributive property.

$f (- 2 x - 3) = 2 (- 2 x (- 2 x) - 2 x \cdot - 3 - 3 (- 2 x - 3)) - 5 (- 2 x - 3) + 10$

Apply the distributive property.

$f (- 2 x - 3) = 2 (- 2 x (- 2 x) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (- 2 x - 3) = 2 (- 2 \cdot (- 2 x \cdot x) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

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

Move $x$ .

$f (- 2 x - 3) = 2 (- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

Multiply $x$ by $x$ .

$f (- 2 x - 3) = 2 (- 2 \cdot (- 2 x^{2}) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

Multiply $- 2$ by $- 2$ .

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

Multiply $- 3$ by $- 2$ .

$f (- 2 x - 3) = 2 (4 x^{2} + 6 x - 3 (- 2 x) - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

Multiply $- 2$ by $- 3$ .

$f (- 2 x - 3) = 2 (4 x^{2} + 6 x + 6 x - 3 \cdot - 3) - 5 (- 2 x - 3) + 10$

Multiply $- 3$ by $- 3$ .

$f (- 2 x - 3) = 2 (4 x^{2} + 6 x + 6 x + 9) - 5 (- 2 x - 3) + 10$

Add $6 x$ and $6 x$ .

$f (- 2 x - 3) = 2 (4 x^{2} + 12 x + 9) - 5 (- 2 x - 3) + 10$

Apply the distributive property.

$f (- 2 x - 3) = 2 (4 x^{2}) + 2 (12 x) + 2 \cdot 9 - 5 (- 2 x - 3) + 10$

Simplify.

Multiply $4$ by $2$ .

$f (- 2 x - 3) = 8 x^{2} + 2 (12 x) + 2 \cdot 9 - 5 (- 2 x - 3) + 10$

Multiply $12$ by $2$ .

$f (- 2 x - 3) = 8 x^{2} + 24 x + 2 \cdot 9 - 5 (- 2 x - 3) + 10$

Multiply $2$ by $9$ .

$f (- 2 x - 3) = 8 x^{2} + 24 x + 18 - 5 (- 2 x - 3) + 10$

Apply the distributive property.

$f (- 2 x - 3) = 8 x^{2} + 24 x + 18 - 5 (- 2 x) - 5 \cdot - 3 + 10$

Multiply $- 2$ by $- 5$ .

$f (- 2 x - 3) = 8 x^{2} + 24 x + 18 + 10 x - 5 \cdot - 3 + 10$

Multiply $- 5$ by $- 3$ .

$f (- 2 x - 3) = 8 x^{2} + 24 x + 18 + 10 x + 15 + 10$

Step 4

Simplify by adding terms.

Add $24 x$ and $10 x$ .

$f (- 2 x - 3) = 8 x^{2} + 34 x + 18 + 15 + 10$

Simplify by adding numbers.

Add $18$ and $15$ .

$f (- 2 x - 3) = 8 x^{2} + 34 x + 33 + 10$

Add $33$ and $10$ .

$f (- 2 x - 3) = 8 x^{2} + 34 x + 43$

Find f(g(x)) f(x)=2x^2-5x+10 g(x)=-2x-3

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