Precalculus Examples

Given that $f (x) = x^{2} + x$ and $g (x) = 3 x + 1$ ; find the value of $f (g (4))$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x + 1)$ by substituting in the value of $g$ into $f$ .

$f (3 x + 1) = {(3 x + 1)}^{2} + 3 x + 1$

Step 3

Remove parentheses.

$f (3 x + 1) = {(3 x + 1)}^{2} + 3 x + 1$

Step 4

Simplify each term.

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Step 4.1

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$f (3 x + 1) = (3 x + 1) (3 x + 1) + 3 x + 1$

Step 4.2

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$f (3 x + 1) = 3 x (3 x + 1) + 1 (3 x + 1) + 3 x + 1$

Step 4.2.2

Apply the distributive property.

$f (3 x + 1) = 3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1) + 3 x + 1$

Step 4.2.3

Apply the distributive property.

$f (3 x + 1) = 3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$f (3 x + 1) = 3 \cdot (3 x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x$ .

$f (3 x + 1) = 3 \cdot (3 (x \cdot x)) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.2.2

Multiply $x$ by $x$ .

$f (3 x + 1) = 3 \cdot (3 x^{2}) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.3

Multiply $3$ by $3$ .

$f (3 x + 1) = 9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.4

Multiply $3$ by $1$ .

$f (3 x + 1) = 9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.5

Multiply $3 x$ by $1$ .

$f (3 x + 1) = 9 x^{2} + 3 x + 3 x + 1 \cdot 1 + 3 x + 1$

Step 4.3.1.6

Multiply $1$ by $1$ .

$f (3 x + 1) = 9 x^{2} + 3 x + 3 x + 1 + 3 x + 1$

Step 4.3.2

Add $3 x$ and $3 x$ .

$f (3 x + 1) = 9 x^{2} + 6 x + 1 + 3 x + 1$

Step 5

Simplify by adding terms.

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Step 5.1

Add $6 x$ and $3 x$ .

$f (3 x + 1) = 9 x^{2} + 9 x + 1 + 1$

Step 5.2

Add $1$ and $1$ .

$f (3 x + 1) = 9 x^{2} + 9 x + 2$

Step 6

Evaluate the result function by replacing $x$ with $4$ in the function.

$9 {(4)}^{2} + 9 (4) + 2$

Step 7

Simplify the evaluated function.

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Step 7.1

Simplify each term.

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Step 7.1.1

Raise $4$ to the power of $2$ .

$9 \cdot 16 + 9 (4) + 2$

Step 7.1.2

Multiply $9$ by $16$ .

$144 + 9 (4) + 2$

Step 7.1.3

Multiply $9$ by $4$ .

$144 + 36 + 2$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add $144$ and $36$ .

$180 + 2$

Step 7.2.2

Add $180$ and $2$ .

$182$