Precalculus Examples

$f (x) = 4 x^{2} - x + 10$ , $g (x) = 2 x - 7$

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 $f$ .

$f (2 x - 7) = 4 {(2 x - 7)}^{2} - (2 x - 7) + 10$

Step 3

Simplify each term.

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Step 3.1

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

$f (2 x - 7) = 4 ((2 x - 7) (2 x - 7)) - (2 x - 7) + 10$

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.

$f (2 x - 7) = 4 (2 x (2 x - 7) - 7 (2 x - 7)) - (2 x - 7) + 10$

Step 3.2.2

Apply the distributive property.

$f (2 x - 7) = 4 (2 x (2 x) + 2 x \cdot - 7 - 7 (2 x - 7)) - (2 x - 7) + 10$

Step 3.2.3

Apply the distributive property.

$f (2 x - 7) = 4 (2 x (2 x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

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.

$f (2 x - 7) = 4 (2 \cdot (2 x \cdot x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$f (2 x - 7) = 4 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$f (2 x - 7) = 4 (2 \cdot (2 x^{2}) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.3

Multiply $2$ by $2$ .

$f (2 x - 7) = 4 (4 x^{2} + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.4

Multiply $- 7$ by $2$ .

$f (2 x - 7) = 4 (4 x^{2} - 14 x - 7 (2 x) - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.5

Multiply $2$ by $- 7$ .

$f (2 x - 7) = 4 (4 x^{2} - 14 x - 14 x - 7 \cdot - 7) - (2 x - 7) + 10$

Step 3.3.1.6

Multiply $- 7$ by $- 7$ .

$f (2 x - 7) = 4 (4 x^{2} - 14 x - 14 x + 49) - (2 x - 7) + 10$

Step 3.3.2

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

$f (2 x - 7) = 4 (4 x^{2} - 28 x + 49) - (2 x - 7) + 10$

Step 3.4

Apply the distributive property.

$f (2 x - 7) = 4 (4 x^{2}) + 4 (- 28 x) + 4 \cdot 49 - (2 x - 7) + 10$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $4$ by $4$ .

$f (2 x - 7) = 16 x^{2} + 4 (- 28 x) + 4 \cdot 49 - (2 x - 7) + 10$

Step 3.5.2

Multiply $- 28$ by $4$ .

$f (2 x - 7) = 16 x^{2} - 112 x + 4 \cdot 49 - (2 x - 7) + 10$

Step 3.5.3

Multiply $4$ by $49$ .

$f (2 x - 7) = 16 x^{2} - 112 x + 196 - (2 x - 7) + 10$

Step 3.6

Apply the distributive property.

$f (2 x - 7) = 16 x^{2} - 112 x + 196 - (2 x) + 7 + 10$

Step 3.7

Multiply $2$ by $- 1$ .

$f (2 x - 7) = 16 x^{2} - 112 x + 196 - 2 x + 7 + 10$

Step 3.8

Multiply $- 1$ by $- 7$ .

$f (2 x - 7) = 16 x^{2} - 112 x + 196 - 2 x + 7 + 10$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $2 x$ from $- 112 x$ .

$f (2 x - 7) = 16 x^{2} - 114 x + 196 + 7 + 10$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $196$ and $7$ .

$f (2 x - 7) = 16 x^{2} - 114 x + 203 + 10$

Step 4.2.2

Add $203$ and $10$ .

$f (2 x - 7) = 16 x^{2} - 114 x + 213$