Precalculus Examples

$f (x) = 6 x - 1$ and $g (x) = 4 x^{2} + x$ ; find $((g \circ f)) (x)$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (6 x - 1)$ by substituting in the value of $f$ into $g$ .

$g (6 x - 1) = 4 {(6 x - 1)}^{2} + 6 x - 1$

Step 3

Remove parentheses.

$g (6 x - 1) = 4 {(6 x - 1)}^{2} + 6 x - 1$

Step 4

Simplify each term.

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Step 4.1

Rewrite ${(6 x - 1)}^{2}$ as $(6 x - 1) (6 x - 1)$ .

$g (6 x - 1) = 4 ((6 x - 1) (6 x - 1)) + 6 x - 1$

Step 4.2

Expand $(6 x - 1) (6 x - 1)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$g (6 x - 1) = 4 (6 x (6 x - 1) - 1 (6 x - 1)) + 6 x - 1$

Step 4.2.2

Apply the distributive property.

$g (6 x - 1) = 4 (6 x (6 x) + 6 x \cdot - 1 - 1 (6 x - 1)) + 6 x - 1$

Step 4.2.3

Apply the distributive property.

$g (6 x - 1) = 4 (6 x (6 x) + 6 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

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

Rewrite using the commutative property of multiplication.

$g (6 x - 1) = 4 (6 \cdot (6 x \cdot x) + 6 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.2.1

Move $x$ .

$g (6 x - 1) = 4 (6 \cdot (6 (x \cdot x)) + 6 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.2.2

Multiply $x$ by $x$ .

$g (6 x - 1) = 4 (6 \cdot (6 x^{2}) + 6 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.3

Multiply $6$ by $6$ .

$g (6 x - 1) = 4 (36 x^{2} + 6 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.4

Multiply $- 1$ by $6$ .

$g (6 x - 1) = 4 (36 x^{2} - 6 x - 1 (6 x) - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.5

Multiply $6$ by $- 1$ .

$g (6 x - 1) = 4 (36 x^{2} - 6 x - 6 x - 1 \cdot - 1) + 6 x - 1$

Step 4.3.1.6

Multiply $- 1$ by $- 1$ .

$g (6 x - 1) = 4 (36 x^{2} - 6 x - 6 x + 1) + 6 x - 1$

Step 4.3.2

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

$g (6 x - 1) = 4 (36 x^{2} - 12 x + 1) + 6 x - 1$

Step 4.4

Apply the distributive property.

$g (6 x - 1) = 4 (36 x^{2}) + 4 (- 12 x) + 4 \cdot 1 + 6 x - 1$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $36$ by $4$ .

$g (6 x - 1) = 144 x^{2} + 4 (- 12 x) + 4 \cdot 1 + 6 x - 1$

Step 4.5.2

Multiply $- 12$ by $4$ .

$g (6 x - 1) = 144 x^{2} - 48 x + 4 \cdot 1 + 6 x - 1$

Step 4.5.3

Multiply $4$ by $1$ .

$g (6 x - 1) = 144 x^{2} - 48 x + 4 + 6 x - 1$

Step 5

Simplify by adding terms.

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Step 5.1

Add $- 48 x$ and $6 x$ .

$g (6 x - 1) = 144 x^{2} - 42 x + 4 - 1$

Step 5.2

Subtract $1$ from $4$ .

$g (6 x - 1) = 144 x^{2} - 42 x + 3$