Algebra Examples

$f (x) = x^{2} + 3$ , $g (x) = \sqrt{x} - 3$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\sqrt{x} - 3)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

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

$f (\sqrt{x} - 3) = (\sqrt{x} - 3) (\sqrt{x} - 3) + 3$

Step 3.2

Expand $(\sqrt{x} - 3) (\sqrt{x} - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$f (\sqrt{x} - 3) = \sqrt{x} (\sqrt{x} - 3) - 3 (\sqrt{x} - 3) + 3$

Step 3.2.2

Apply the distributive property.

$f (\sqrt{x} - 3) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 3 - 3 (\sqrt{x} - 3) + 3$

Step 3.2.3

Apply the distributive property.

$f (\sqrt{x} - 3) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 3 - 3 \sqrt{x} - 3 \cdot - 3 + 3$

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

Multiply $\sqrt{x} \sqrt{x}$ .

Tap for more steps...

Step 3.3.1.1.1

Raise $\sqrt{x}$ to the power of $1$ .

$f (\sqrt{x} - 3) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 3 - 3 \sqrt{x} - 3 \cdot - 3 + 3$

Step 3.3.1.1.2

Raise $\sqrt{x}$ to the power of $1$ .

$f (\sqrt{x} - 3) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 3 - 3 \sqrt{x} - 3 \cdot - 3 + 3$

Step 3.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.1.1.4

Add $1$ and $1$ .

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

Step 3.3.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Tap for more steps...

Step 3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.2.4.1

Cancel the common factor.

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

Step 3.3.1.2.4.2

Rewrite the expression.

$f (\sqrt{x} - 3) = x + \sqrt{x} \cdot - 3 - 3 \sqrt{x} - 3 \cdot - 3 + 3$

Step 3.3.1.2.5

Simplify.

$f (\sqrt{x} - 3) = x + \sqrt{x} \cdot - 3 - 3 \sqrt{x} - 3 \cdot - 3 + 3$

Step 3.3.1.3

Move $- 3$ to the left of $\sqrt{x}$ .

$f (\sqrt{x} - 3) = x - 3 \cdot \sqrt{x} - 3 \sqrt{x} - 3 \cdot - 3 + 3$

Step 3.3.1.4

Multiply $- 3$ by $- 3$ .

$f (\sqrt{x} - 3) = x - 3 \sqrt{x} - 3 \sqrt{x} + 9 + 3$

Step 3.3.2

Subtract $3 \sqrt{x}$ from $- 3 \sqrt{x}$ .

$f (\sqrt{x} - 3) = x - 6 \sqrt{x} + 9 + 3$

Step 4

Add $9$ and $3$ .

$f (\sqrt{x} - 3) = x - 6 \sqrt{x} + 12$