Precalculus Examples

If $f (x) = x - 4$ and $g (x) = 3 x^{2} - x$ ; what is $((g \circ f)) (x)$ ?

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x - 4)$ by substituting in the value of $f$ into $g$ .

$g (x - 4) = 3 {(x - 4)}^{2} - (x - 4)$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$g (x - 4) = 3 ((x - 4) (x - 4)) - (x - 4)$

Step 3.2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$g (x - 4) = 3 (x (x - 4) - 4 (x - 4)) - (x - 4)$

Step 3.2.2

Apply the distributive property.

$g (x - 4) = 3 (x \cdot x + x \cdot - 4 - 4 (x - 4)) - (x - 4)$

Step 3.2.3

Apply the distributive property.

$g (x - 4) = 3 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) - (x - 4)$

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 $x$ .

$g (x - 4) = 3 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) - (x - 4)$

Step 3.3.1.2

Move $- 4$ to the left of $x$ .

$g (x - 4) = 3 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) - (x - 4)$

Step 3.3.1.3

Multiply $- 4$ by $- 4$ .

$g (x - 4) = 3 (x^{2} - 4 x - 4 x + 16) - (x - 4)$

Step 3.3.2

Subtract $4 x$ from $- 4 x$ .

$g (x - 4) = 3 (x^{2} - 8 x + 16) - (x - 4)$

Step 3.4

Apply the distributive property.

$g (x - 4) = 3 x^{2} + 3 (- 8 x) + 3 \cdot 16 - (x - 4)$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $- 8$ by $3$ .

$g (x - 4) = 3 x^{2} - 24 x + 3 \cdot 16 - (x - 4)$

Step 3.5.2

Multiply $3$ by $16$ .

$g (x - 4) = 3 x^{2} - 24 x + 48 - (x - 4)$

Step 3.6

Apply the distributive property.

$g (x - 4) = 3 x^{2} - 24 x + 48 - x + 4$

Step 3.7

Multiply $- 1$ by $- 4$ .

$g (x - 4) = 3 x^{2} - 24 x + 48 - x + 4$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $x$ from $- 24 x$ .

$g (x - 4) = 3 x^{2} - 25 x + 48 + 4$

Step 4.2

Add $48$ and $4$ .

$g (x - 4) = 3 x^{2} - 25 x + 52$