Precalculus Examples

$f (x) = 2 x + 1$ $g (x) = x^{2} + 5 x + 10$ Find: $g (f (x))$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (2 x + 1)$ by substituting in the value of $f$ into $g$ .

$g (2 x + 1) = {(2 x + 1)}^{2} + 5 (2 x + 1) + 10$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(2 x + 1)}^{2}$ as $(2 x + 1) (2 x + 1)$ .

$g (2 x + 1) = (2 x + 1) (2 x + 1) + 5 (2 x + 1) + 10$

Step 3.2

Expand $(2 x + 1) (2 x + 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$g (2 x + 1) = 2 x (2 x + 1) + 1 (2 x + 1) + 5 (2 x + 1) + 10$

Step 3.2.2

Apply the distributive property.

$g (2 x + 1) = 2 x (2 x) + 2 x \cdot 1 + 1 (2 x + 1) + 5 (2 x + 1) + 10$

Step 3.2.3

Apply the distributive property.

$g (2 x + 1) = 2 x (2 x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$g (2 x + 1) = 2 \cdot (2 x \cdot x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$g (2 x + 1) = 2 \cdot (2 (x \cdot x)) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$g (2 x + 1) = 2 \cdot (2 x^{2}) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.3

Multiply $2$ by $2$ .

$g (2 x + 1) = 4 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.4

Multiply $2$ by $1$ .

$g (2 x + 1) = 4 x^{2} + 2 x + 1 (2 x) + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.5

Multiply $2 x$ by $1$ .

$g (2 x + 1) = 4 x^{2} + 2 x + 2 x + 1 \cdot 1 + 5 (2 x + 1) + 10$

Step 3.3.1.6

Multiply $1$ by $1$ .

$g (2 x + 1) = 4 x^{2} + 2 x + 2 x + 1 + 5 (2 x + 1) + 10$

Step 3.3.2

Add $2 x$ and $2 x$ .

$g (2 x + 1) = 4 x^{2} + 4 x + 1 + 5 (2 x + 1) + 10$

Step 3.4

Apply the distributive property.

$g (2 x + 1) = 4 x^{2} + 4 x + 1 + 5 (2 x) + 5 \cdot 1 + 10$

Step 3.5

Multiply $2$ by $5$ .

$g (2 x + 1) = 4 x^{2} + 4 x + 1 + 10 x + 5 \cdot 1 + 10$

Step 3.6

Multiply $5$ by $1$ .

$g (2 x + 1) = 4 x^{2} + 4 x + 1 + 10 x + 5 + 10$

Step 4

Simplify by adding terms.

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Step 4.1

Add $4 x$ and $10 x$ .

$g (2 x + 1) = 4 x^{2} + 14 x + 1 + 5 + 10$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $1$ and $5$ .

$g (2 x + 1) = 4 x^{2} + 14 x + 6 + 10$

Step 4.2.2

Add $6$ and $10$ .

$g (2 x + 1) = 4 x^{2} + 14 x + 16$