Precalculus Examples

$f (x) = x - 8$ $g (x) = x^{2} + x - 15$ Find: $((g \circ f)) (x)$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x - 8)$ by substituting in the value of $f$ into $g$ .

$g (x - 8) = {(x - 8)}^{2} + x - 8 - 15$

Step 3

Remove parentheses.

$g (x - 8) = {(x - 8)}^{2} + x - 8 - 15$

Step 4

Simplify each term.

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Step 4.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$g (x - 8) = (x - 8) (x - 8) + x - 8 - 15$

Step 4.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$g (x - 8) = x (x - 8) - 8 (x - 8) + x - 8 - 15$

Step 4.2.2

Apply the distributive property.

$g (x - 8) = x \cdot x + x \cdot - 8 - 8 (x - 8) + x - 8 - 15$

Step 4.2.3

Apply the distributive property.

$g (x - 8) = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + x - 8 - 15$

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

Multiply $x$ by $x$ .

$g (x - 8) = x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 + x - 8 - 15$

Step 4.3.1.2

Move $- 8$ to the left of $x$ .

$g (x - 8) = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 + x - 8 - 15$

Step 4.3.1.3

Multiply $- 8$ by $- 8$ .

$g (x - 8) = x^{2} - 8 x - 8 x + 64 + x - 8 - 15$

Step 4.3.2

Subtract $8 x$ from $- 8 x$ .

$g (x - 8) = x^{2} - 16 x + 64 + x - 8 - 15$

Step 5

Simplify by adding terms.

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Step 5.1

Add $- 16 x$ and $x$ .

$g (x - 8) = x^{2} - 15 x + 64 - 8 - 15$

Step 5.2

Simplify by subtracting numbers.

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Step 5.2.1

Subtract $8$ from $64$ .

$g (x - 8) = x^{2} - 15 x + 56 - 15$

Step 5.2.2

Subtract $15$ from $56$ .

$g (x - 8) = x^{2} - 15 x + 41$