Precalculus Examples

$f (x) = x^{2} - 5 x - 10$ , $g (x) = - 7 x + 14$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (- 7 x + 14)$ by substituting in the value of $g$ into $f$ .

$f (- 7 x + 14) = {(- 7 x + 14)}^{2} - 5 (- 7 x + 14) - 10$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(- 7 x + 14)}^{2}$ as $(- 7 x + 14) (- 7 x + 14)$ .

$f (- 7 x + 14) = (- 7 x + 14) (- 7 x + 14) - 5 (- 7 x + 14) - 10$

Step 3.2

Expand $(- 7 x + 14) (- 7 x + 14)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (- 7 x + 14) = - 7 x (- 7 x + 14) + 14 (- 7 x + 14) - 5 (- 7 x + 14) - 10$

Step 3.2.2

Apply the distributive property.

$f (- 7 x + 14) = - 7 x (- 7 x) - 7 x \cdot 14 + 14 (- 7 x + 14) - 5 (- 7 x + 14) - 10$

Step 3.2.3

Apply the distributive property.

$f (- 7 x + 14) = - 7 x (- 7 x) - 7 x \cdot 14 + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

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.

$f (- 7 x + 14) = - 7 \cdot (- 7 x \cdot x) - 7 x \cdot 14 + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$f (- 7 x + 14) = - 7 \cdot (- 7 (x \cdot x)) - 7 x \cdot 14 + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$f (- 7 x + 14) = - 7 \cdot (- 7 x^{2}) - 7 x \cdot 14 + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.3

Multiply $- 7$ by $- 7$ .

$f (- 7 x + 14) = 49 x^{2} - 7 x \cdot 14 + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.4

Multiply $14$ by $- 7$ .

$f (- 7 x + 14) = 49 x^{2} - 98 x + 14 (- 7 x) + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.5

Multiply $- 7$ by $14$ .

$f (- 7 x + 14) = 49 x^{2} - 98 x - 98 x + 14 \cdot 14 - 5 (- 7 x + 14) - 10$

Step 3.3.1.6

Multiply $14$ by $14$ .

$f (- 7 x + 14) = 49 x^{2} - 98 x - 98 x + 196 - 5 (- 7 x + 14) - 10$

Step 3.3.2

Subtract $98 x$ from $- 98 x$ .

$f (- 7 x + 14) = 49 x^{2} - 196 x + 196 - 5 (- 7 x + 14) - 10$

Step 3.4

Apply the distributive property.

$f (- 7 x + 14) = 49 x^{2} - 196 x + 196 - 5 (- 7 x) - 5 \cdot 14 - 10$

Step 3.5

Multiply $- 7$ by $- 5$ .

$f (- 7 x + 14) = 49 x^{2} - 196 x + 196 + 35 x - 5 \cdot 14 - 10$

Step 3.6

Multiply $- 5$ by $14$ .

$f (- 7 x + 14) = 49 x^{2} - 196 x + 196 + 35 x - 70 - 10$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 196 x$ and $35 x$ .

$f (- 7 x + 14) = 49 x^{2} - 161 x + 196 - 70 - 10$

Step 4.2

Simplify by subtracting numbers.

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Step 4.2.1

Subtract $70$ from $196$ .

$f (- 7 x + 14) = 49 x^{2} - 161 x + 126 - 10$

Step 4.2.2

Subtract $10$ from $126$ .

$f (- 7 x + 14) = 49 x^{2} - 161 x + 116$