Precalculus Examples

$f (x) = 2 x^{2} - 2 x + 4$ $g (x) = x + 6$ Find: $((f \circ g)) (x)$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x + 6)$ by substituting in the value of $g$ into $f$ .

$f (x + 6) = 2 {(x + 6)}^{2} - 2 (x + 6) + 4$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$f (x + 6) = 2 ((x + 6) (x + 6)) - 2 (x + 6) + 4$

Step 3.2

Expand $(x + 6) (x + 6)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (x + 6) = 2 (x (x + 6) + 6 (x + 6)) - 2 (x + 6) + 4$

Step 3.2.2

Apply the distributive property.

$f (x + 6) = 2 (x \cdot x + x \cdot 6 + 6 (x + 6)) - 2 (x + 6) + 4$

Step 3.2.3

Apply the distributive property.

$f (x + 6) = 2 (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6) - 2 (x + 6) + 4$

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 $x$ .

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

Step 3.3.1.2

Move $6$ to the left of $x$ .

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

Step 3.3.1.3

Multiply $6$ by $6$ .

$f (x + 6) = 2 (x^{2} + 6 x + 6 x + 36) - 2 (x + 6) + 4$

Step 3.3.2

Add $6 x$ and $6 x$ .

$f (x + 6) = 2 (x^{2} + 12 x + 36) - 2 (x + 6) + 4$

Step 3.4

Apply the distributive property.

$f (x + 6) = 2 x^{2} + 2 (12 x) + 2 \cdot 36 - 2 (x + 6) + 4$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $12$ by $2$ .

$f (x + 6) = 2 x^{2} + 24 x + 2 \cdot 36 - 2 (x + 6) + 4$

Step 3.5.2

Multiply $2$ by $36$ .

$f (x + 6) = 2 x^{2} + 24 x + 72 - 2 (x + 6) + 4$

Step 3.6

Apply the distributive property.

$f (x + 6) = 2 x^{2} + 24 x + 72 - 2 x - 2 \cdot 6 + 4$

Step 3.7

Multiply $- 2$ by $6$ .

$f (x + 6) = 2 x^{2} + 24 x + 72 - 2 x - 12 + 4$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $2 x$ from $24 x$ .

$f (x + 6) = 2 x^{2} + 22 x + 72 - 12 + 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $12$ from $72$ .

$f (x + 6) = 2 x^{2} + 22 x + 60 + 4$

Step 4.2.2

Add $60$ and $4$ .

$f (x + 6) = 2 x^{2} + 22 x + 64$