Precalculus Examples

$f (x) = x + 1$ $g (x) = 2 x^{2} - 2 x - 1$ Find: $((g \circ f)) (x)$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

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

$g (x + 1) = 2 ((x + 1) (x + 1)) - 2 (x + 1) - 1$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$g (x + 1) = 2 (x (x + 1) + 1 (x + 1)) - 2 (x + 1) - 1$

Step 3.2.2

Apply the distributive property.

$g (x + 1) = 2 (x \cdot x + x \cdot 1 + 1 (x + 1)) - 2 (x + 1) - 1$

Step 3.2.3

Apply the distributive property.

$g (x + 1) = 2 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) - 2 (x + 1) - 1$

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

Multiply $x$ by $x$ .

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

Step 3.3.1.2

Multiply $x$ by $1$ .

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

Step 3.3.1.3

Multiply $x$ by $1$ .

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

Step 3.3.1.4

Multiply $1$ by $1$ .

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

Step 3.3.2

Add $x$ and $x$ .

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Simplify.

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Step 3.5.1

Multiply $2$ by $2$ .

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

Step 3.5.2

Multiply $2$ by $1$ .

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Multiply $- 2$ by $1$ .

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

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $2 x^{2} + 4 x + 2 - 2 x - 2 - 1$ .

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Step 4.1.1

Subtract $2$ from $2$ .

$g (x + 1) = 2 x^{2} + 4 x - 2 x + 0 - 1$

Step 4.1.2

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

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

Step 4.2

Subtract $2 x$ from $4 x$ .

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