Algebra Examples

$f (x) = x - 6$ , $g (x) = x^{\frac{1}{2}} (x + 3)$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{\frac{1}{2}} (x + 3))$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{1}{2}} x + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.2

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 3.2.1

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 3.2.1.1

Raise $x$ to the power of $1$ .

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{1}{2}} x + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.2.1.2

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

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{1}{2} + 1} + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.2.2

Write $1$ as a fraction with a common denominator.

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{1}{2} + \frac{2}{2}} + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.2.3

Combine the numerators over the common denominator.

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{1 + 2}{2}} + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.2.4

Add $1$ and $2$ .

$f (x^{\frac{1}{2}} (x + 3)) = x^{\frac{3}{2}} + x^{\frac{1}{2}} \cdot 3 - 6$

Step 3.3

Move $3$ to the left of $x^{\frac{1}{2}}$ .

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