Precalculus Examples

Find $(f \circ g)$ if $f (x) = 4 x^{2} - 6 x + 3$ and $g (x) = x - 6$

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 $f$ .

$f (x - 6) = 4 {(x - 6)}^{2} - 6 (x - 6) + 3$

Step 3

Simplify each term.

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Step 3.1

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

$f (x - 6) = 4 ((x - 6) (x - 6)) - 6 (x - 6) + 3$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$f (x - 6) = 4 (x (x - 6) - 6 (x - 6)) - 6 (x - 6) + 3$

Step 3.2.2

Apply the distributive property.

$f (x - 6) = 4 (x \cdot x + x \cdot - 6 - 6 (x - 6)) - 6 (x - 6) + 3$

Step 3.2.3

Apply the distributive property.

$f (x - 6) = 4 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - 6 (x - 6) + 3$

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 $x$ .

$f (x - 6) = 4 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) - 6 (x - 6) + 3$

Step 3.3.1.2

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

$f (x - 6) = 4 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) - 6 (x - 6) + 3$

Step 3.3.1.3

Multiply $- 6$ by $- 6$ .

$f (x - 6) = 4 (x^{2} - 6 x - 6 x + 36) - 6 (x - 6) + 3$

Step 3.3.2

Subtract $6 x$ from $- 6 x$ .

$f (x - 6) = 4 (x^{2} - 12 x + 36) - 6 (x - 6) + 3$

Step 3.4

Apply the distributive property.

$f (x - 6) = 4 x^{2} + 4 (- 12 x) + 4 \cdot 36 - 6 (x - 6) + 3$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $- 12$ by $4$ .

$f (x - 6) = 4 x^{2} - 48 x + 4 \cdot 36 - 6 (x - 6) + 3$

Step 3.5.2

Multiply $4$ by $36$ .

$f (x - 6) = 4 x^{2} - 48 x + 144 - 6 (x - 6) + 3$

Step 3.6

Apply the distributive property.

$f (x - 6) = 4 x^{2} - 48 x + 144 - 6 x - 6 \cdot - 6 + 3$

Step 3.7

Multiply $- 6$ by $- 6$ .

$f (x - 6) = 4 x^{2} - 48 x + 144 - 6 x + 36 + 3$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $6 x$ from $- 48 x$ .

$f (x - 6) = 4 x^{2} - 54 x + 144 + 36 + 3$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $144$ and $36$ .

$f (x - 6) = 4 x^{2} - 54 x + 180 + 3$

Step 4.2.2

Add $180$ and $3$ .

$f (x - 6) = 4 x^{2} - 54 x + 183$