Precalculus Examples

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (- 2 x + 5)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Expand $(- 2 x + 5) (- 2 x + 5)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x$ .

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

Step 4.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 4.3.1.4

Multiply $5$ by $- 2$ .

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

Step 4.3.1.5

Multiply $- 2$ by $5$ .

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

Step 4.3.1.6

Multiply $5$ by $5$ .

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

Step 4.3.2

Subtract $10 x$ from $- 10 x$ .

$g (- 2 x + 5) = 4 x^{2} - 20 x + 25 - 2 x + 5 + 10$

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $2 x$ from $- 20 x$ .

$g (- 2 x + 5) = 4 x^{2} - 22 x + 25 + 5 + 10$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $25$ and $5$ .

$g (- 2 x + 5) = 4 x^{2} - 22 x + 30 + 10$

Step 5.2.2

Add $30$ and $10$ .

$g (- 2 x + 5) = 4 x^{2} - 22 x + 40$