Precalculus Examples

$f (x) = 3 x - 10$ $g (x) = x^{2} + 4 x + 5$ Find: $g (f (x))$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x - 10)$ by substituting in the value of $f$ into $g$ .

$g (3 x - 10) = {(3 x - 10)}^{2} + 4 (3 x - 10) + 5$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Rewrite ${(3 x - 10)}^{2}$ as $(3 x - 10) (3 x - 10)$ .

$g (3 x - 10) = (3 x - 10) (3 x - 10) + 4 (3 x - 10) + 5$

Step 3.2

Expand $(3 x - 10) (3 x - 10)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$g (3 x - 10) = 3 x (3 x - 10) - 10 (3 x - 10) + 4 (3 x - 10) + 5$

Step 3.2.2

Apply the distributive property.

$g (3 x - 10) = 3 x (3 x) + 3 x \cdot - 10 - 10 (3 x - 10) + 4 (3 x - 10) + 5$

Step 3.2.3

Apply the distributive property.

$g (3 x - 10) = 3 x (3 x) + 3 x \cdot - 10 - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$g (3 x - 10) = 3 \cdot (3 x \cdot x) + 3 x \cdot - 10 - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.2

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

Tap for more steps...

Step 3.3.1.2.1

Move $x$ .

$g (3 x - 10) = 3 \cdot (3 (x \cdot x)) + 3 x \cdot - 10 - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$g (3 x - 10) = 3 \cdot (3 x^{2}) + 3 x \cdot - 10 - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.3

Multiply $3$ by $3$ .

$g (3 x - 10) = 9 x^{2} + 3 x \cdot - 10 - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.4

Multiply $- 10$ by $3$ .

$g (3 x - 10) = 9 x^{2} - 30 x - 10 (3 x) - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.5

Multiply $3$ by $- 10$ .

$g (3 x - 10) = 9 x^{2} - 30 x - 30 x - 10 \cdot - 10 + 4 (3 x - 10) + 5$

Step 3.3.1.6

Multiply $- 10$ by $- 10$ .

$g (3 x - 10) = 9 x^{2} - 30 x - 30 x + 100 + 4 (3 x - 10) + 5$

Step 3.3.2

Subtract $30 x$ from $- 30 x$ .

$g (3 x - 10) = 9 x^{2} - 60 x + 100 + 4 (3 x - 10) + 5$

Step 3.4

Apply the distributive property.

$g (3 x - 10) = 9 x^{2} - 60 x + 100 + 4 (3 x) + 4 \cdot - 10 + 5$

Step 3.5

Multiply $3$ by $4$ .

$g (3 x - 10) = 9 x^{2} - 60 x + 100 + 12 x + 4 \cdot - 10 + 5$

Step 3.6

Multiply $4$ by $- 10$ .

$g (3 x - 10) = 9 x^{2} - 60 x + 100 + 12 x - 40 + 5$

Step 4

Simplify by adding terms.

Tap for more steps...

Step 4.1

Add $- 60 x$ and $12 x$ .

$g (3 x - 10) = 9 x^{2} - 48 x + 100 - 40 + 5$

Step 4.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.1

Subtract $40$ from $100$ .

$g (3 x - 10) = 9 x^{2} - 48 x + 60 + 5$

Step 4.2.2

Add $60$ and $5$ .

$g (3 x - 10) = 9 x^{2} - 48 x + 65$