Precalculus Examples

$f (x) = 2 x^{2} - 7 x - 4$ $g (x) = 2 x - 4$ Find: $f (g (x))$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (2 x - 4)$ by substituting in the value of $g$ into $f$ .

$f (2 x - 4) = 2 {(2 x - 4)}^{2} - 7 (2 x - 4) - 4$

Step 3

Simplify each term.

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Step 3.1

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

$f (2 x - 4) = 2 ((2 x - 4) (2 x - 4)) - 7 (2 x - 4) - 4$

Step 3.2

Expand $(2 x - 4) (2 x - 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (2 x - 4) = 2 (2 x (2 x - 4) - 4 (2 x - 4)) - 7 (2 x - 4) - 4$

Step 3.2.2

Apply the distributive property.

$f (2 x - 4) = 2 (2 x (2 x) + 2 x \cdot - 4 - 4 (2 x - 4)) - 7 (2 x - 4) - 4$

Step 3.2.3

Apply the distributive property.

$f (2 x - 4) = 2 (2 x (2 x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

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

Rewrite using the commutative property of multiplication.

$f (2 x - 4) = 2 (2 \cdot (2 x \cdot x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$f (2 x - 4) = 2 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$f (2 x - 4) = 2 (2 \cdot (2 x^{2}) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.3

Multiply $2$ by $2$ .

$f (2 x - 4) = 2 (4 x^{2} + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.4

Multiply $- 4$ by $2$ .

$f (2 x - 4) = 2 (4 x^{2} - 8 x - 4 (2 x) - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.5

Multiply $2$ by $- 4$ .

$f (2 x - 4) = 2 (4 x^{2} - 8 x - 8 x - 4 \cdot - 4) - 7 (2 x - 4) - 4$

Step 3.3.1.6

Multiply $- 4$ by $- 4$ .

$f (2 x - 4) = 2 (4 x^{2} - 8 x - 8 x + 16) - 7 (2 x - 4) - 4$

Step 3.3.2

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

$f (2 x - 4) = 2 (4 x^{2} - 16 x + 16) - 7 (2 x - 4) - 4$

Step 3.4

Apply the distributive property.

$f (2 x - 4) = 2 (4 x^{2}) + 2 (- 16 x) + 2 \cdot 16 - 7 (2 x - 4) - 4$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $4$ by $2$ .

$f (2 x - 4) = 8 x^{2} + 2 (- 16 x) + 2 \cdot 16 - 7 (2 x - 4) - 4$

Step 3.5.2

Multiply $- 16$ by $2$ .

$f (2 x - 4) = 8 x^{2} - 32 x + 2 \cdot 16 - 7 (2 x - 4) - 4$

Step 3.5.3

Multiply $2$ by $16$ .

$f (2 x - 4) = 8 x^{2} - 32 x + 32 - 7 (2 x - 4) - 4$

Step 3.6

Apply the distributive property.

$f (2 x - 4) = 8 x^{2} - 32 x + 32 - 7 (2 x) - 7 \cdot - 4 - 4$

Step 3.7

Multiply $2$ by $- 7$ .

$f (2 x - 4) = 8 x^{2} - 32 x + 32 - 14 x - 7 \cdot - 4 - 4$

Step 3.8

Multiply $- 7$ by $- 4$ .

$f (2 x - 4) = 8 x^{2} - 32 x + 32 - 14 x + 28 - 4$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $14 x$ from $- 32 x$ .

$f (2 x - 4) = 8 x^{2} - 46 x + 32 + 28 - 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $32$ and $28$ .

$f (2 x - 4) = 8 x^{2} - 46 x + 60 - 4$

Step 4.2.2

Subtract $4$ from $60$ .

$f (2 x - 4) = 8 x^{2} - 46 x + 56$