Precalculus Examples

If $f (x) = x^{2}$ and $g (x) = 3 x - 1$ ; find $f (g (x))$

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 $f$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.2

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

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Step 5.1.2.1

Move $x$ .

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

Step 5.1.2.2

Multiply $x$ by $x$ .

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

Step 5.1.3

Multiply $3$ by $3$ .

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

Step 5.1.4

Multiply $- 1$ by $3$ .

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

Step 5.1.5

Multiply $3$ by $- 1$ .

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

Step 5.1.6

Multiply $- 1$ by $- 1$ .

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

Step 5.2

Subtract $3 x$ from $- 3 x$ .

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