Precalculus Examples

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

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

$g (- 2 x + 2) = (- 2 x + 2) (- 2 x + 2) - 5 (- 2 x + 2) + 10$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$g (- 2 x + 2) = - 2 x (- 2 x + 2) + 2 (- 2 x + 2) - 5 (- 2 x + 2) + 10$

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.2

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

Tap for more steps...

Step 3.3.1.2.1

Move $x$ .

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

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.3.1.4

Multiply $2$ by $- 2$ .

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

Step 3.3.1.5

Multiply $- 2$ by $2$ .

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

Step 3.3.1.6

Multiply $2$ by $2$ .

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

Step 3.3.2

Subtract $4 x$ from $- 4 x$ .

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 - 5 (- 2 x + 2) + 10$

Step 3.4

Apply the distributive property.

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 - 5 (- 2 x) - 5 \cdot 2 + 10$

Step 3.5

Multiply $- 2$ by $- 5$ .

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 + 10 x - 5 \cdot 2 + 10$

Step 3.6

Multiply $- 5$ by $2$ .

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 + 10 x - 10 + 10$

Step 4

Simplify by adding terms.

Tap for more steps...

Step 4.1

Combine the opposite terms in $4 x^{2} - 8 x + 4 + 10 x - 10 + 10$ .

Tap for more steps...

Step 4.1.1

Add $- 10$ and $10$ .

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 + 10 x + 0$

Step 4.1.2

Add $4 x^{2} - 8 x + 4 + 10 x$ and $0$ .

$g (- 2 x + 2) = 4 x^{2} - 8 x + 4 + 10 x$

Step 4.2

Add $- 8 x$ and $10 x$ .

$g (- 2 x + 2) = 4 x^{2} + 2 x + 4$