Algebra Examples

$f (x) = 4 x^{2} + 3 x + 5$ , $g (x) = 3 x - 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x - 7)$ by substituting in the value of $g$ into $f$ .

$f (3 x - 7) = 4 {(3 x - 7)}^{2} + 3 (3 x - 7) + 5$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(3 x - 7)}^{2}$ as $(3 x - 7) (3 x - 7)$ .

$f (3 x - 7) = 4 ((3 x - 7) (3 x - 7)) + 3 (3 x - 7) + 5$

Step 3.2

Expand $(3 x - 7) (3 x - 7)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (3 x - 7) = 4 (3 x (3 x - 7) - 7 (3 x - 7)) + 3 (3 x - 7) + 5$

Step 3.2.2

Apply the distributive property.

$f (3 x - 7) = 4 (3 x (3 x) + 3 x \cdot - 7 - 7 (3 x - 7)) + 3 (3 x - 7) + 5$

Step 3.2.3

Apply the distributive property.

$f (3 x - 7) = 4 (3 x (3 x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

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.

$f (3 x - 7) = 4 (3 \cdot (3 x \cdot x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$f (3 x - 7) = 4 (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$f (3 x - 7) = 4 (3 \cdot (3 x^{2}) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.3

Multiply $3$ by $3$ .

$f (3 x - 7) = 4 (9 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.4

Multiply $- 7$ by $3$ .

$f (3 x - 7) = 4 (9 x^{2} - 21 x - 7 (3 x) - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.5

Multiply $3$ by $- 7$ .

$f (3 x - 7) = 4 (9 x^{2} - 21 x - 21 x - 7 \cdot - 7) + 3 (3 x - 7) + 5$

Step 3.3.1.6

Multiply $- 7$ by $- 7$ .

$f (3 x - 7) = 4 (9 x^{2} - 21 x - 21 x + 49) + 3 (3 x - 7) + 5$

Step 3.3.2

Subtract $21 x$ from $- 21 x$ .

$f (3 x - 7) = 4 (9 x^{2} - 42 x + 49) + 3 (3 x - 7) + 5$

Step 3.4

Apply the distributive property.

$f (3 x - 7) = 4 (9 x^{2}) + 4 (- 42 x) + 4 \cdot 49 + 3 (3 x - 7) + 5$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $9$ by $4$ .

$f (3 x - 7) = 36 x^{2} + 4 (- 42 x) + 4 \cdot 49 + 3 (3 x - 7) + 5$

Step 3.5.2

Multiply $- 42$ by $4$ .

$f (3 x - 7) = 36 x^{2} - 168 x + 4 \cdot 49 + 3 (3 x - 7) + 5$

Step 3.5.3

Multiply $4$ by $49$ .

$f (3 x - 7) = 36 x^{2} - 168 x + 196 + 3 (3 x - 7) + 5$

Step 3.6

Apply the distributive property.

$f (3 x - 7) = 36 x^{2} - 168 x + 196 + 3 (3 x) + 3 \cdot - 7 + 5$

Step 3.7

Multiply $3$ by $3$ .

$f (3 x - 7) = 36 x^{2} - 168 x + 196 + 9 x + 3 \cdot - 7 + 5$

Step 3.8

Multiply $3$ by $- 7$ .

$f (3 x - 7) = 36 x^{2} - 168 x + 196 + 9 x - 21 + 5$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 168 x$ and $9 x$ .

$f (3 x - 7) = 36 x^{2} - 159 x + 196 - 21 + 5$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $21$ from $196$ .

$f (3 x - 7) = 36 x^{2} - 159 x + 175 + 5$

Step 4.2.2

Add $175$ and $5$ .

$f (3 x - 7) = 36 x^{2} - 159 x + 180$