Precalculus Examples

$f (x) = x^{2} + 4 x - 13$ $g (x) = - 2 x + 7$ Find: $f (g (x))$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (- 2 x + 7)$ by substituting in the value of $g$ into $f$ .

$f (- 2 x + 7) = {(- 2 x + 7)}^{2} + 4 (- 2 x + 7) - 13$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(- 2 x + 7)}^{2}$ as $(- 2 x + 7) (- 2 x + 7)$ .

$f (- 2 x + 7) = (- 2 x + 7) (- 2 x + 7) + 4 (- 2 x + 7) - 13$

Step 3.2

Expand $(- 2 x + 7) (- 2 x + 7)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (- 2 x + 7) = - 2 x (- 2 x + 7) + 7 (- 2 x + 7) + 4 (- 2 x + 7) - 13$

Step 3.2.2

Apply the distributive property.

$f (- 2 x + 7) = - 2 x (- 2 x) - 2 x \cdot 7 + 7 (- 2 x + 7) + 4 (- 2 x + 7) - 13$

Step 3.2.3

Apply the distributive property.

$f (- 2 x + 7) = - 2 x (- 2 x) - 2 x \cdot 7 + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

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.

$f (- 2 x + 7) = - 2 \cdot (- 2 x \cdot x) - 2 x \cdot 7 + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$f (- 2 x + 7) = - 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot 7 + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$f (- 2 x + 7) = - 2 \cdot (- 2 x^{2}) - 2 x \cdot 7 + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.3

Multiply $- 2$ by $- 2$ .

$f (- 2 x + 7) = 4 x^{2} - 2 x \cdot 7 + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.4

Multiply $7$ by $- 2$ .

$f (- 2 x + 7) = 4 x^{2} - 14 x + 7 (- 2 x) + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.5

Multiply $- 2$ by $7$ .

$f (- 2 x + 7) = 4 x^{2} - 14 x - 14 x + 7 \cdot 7 + 4 (- 2 x + 7) - 13$

Step 3.3.1.6

Multiply $7$ by $7$ .

$f (- 2 x + 7) = 4 x^{2} - 14 x - 14 x + 49 + 4 (- 2 x + 7) - 13$

Step 3.3.2

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

$f (- 2 x + 7) = 4 x^{2} - 28 x + 49 + 4 (- 2 x + 7) - 13$

Step 3.4

Apply the distributive property.

$f (- 2 x + 7) = 4 x^{2} - 28 x + 49 + 4 (- 2 x) + 4 \cdot 7 - 13$

Step 3.5

Multiply $- 2$ by $4$ .

$f (- 2 x + 7) = 4 x^{2} - 28 x + 49 - 8 x + 4 \cdot 7 - 13$

Step 3.6

Multiply $4$ by $7$ .

$f (- 2 x + 7) = 4 x^{2} - 28 x + 49 - 8 x + 28 - 13$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $8 x$ from $- 28 x$ .

$f (- 2 x + 7) = 4 x^{2} - 36 x + 49 + 28 - 13$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $49$ and $28$ .

$f (- 2 x + 7) = 4 x^{2} - 36 x + 77 - 13$

Step 4.2.2

Subtract $13$ from $77$ .

$f (- 2 x + 7) = 4 x^{2} - 36 x + 64$