Algebra Examples

$f (x) = \sqrt{2 x}$ , $g (x) = \sqrt{8 x} - \sqrt{32}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\sqrt{8 x} - \sqrt{32})$ by substituting in the value of $g$ into $f$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (\sqrt{8 x} - \sqrt{32})}$

Step 3

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 3.1

Factor $4$ out of $8$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (\sqrt{4 (2) x} - \sqrt{32})}$

Step 3.2

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (\sqrt{2^{2} \cdot (2 x)} - \sqrt{32})}$

Step 3.3

Add parentheses.

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (\sqrt{2^{2} \cdot (2 x)} - \sqrt{32})}$

Step 4

Pull terms out from under the radical.

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 \sqrt{2 x} - \sqrt{32})}$

Step 5

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.1

Factor $16$ out of $32$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 \sqrt{2 x} - \sqrt{16 (2)})}$

Step 5.2

Rewrite $16$ as $4^{2}$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 \sqrt{2 x} - \sqrt{4^{2} \cdot 2})}$

Step 6

Pull terms out from under the radical.

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 \sqrt{2 x} - (4 \sqrt{2}))}$

Step 7

Multiply $4$ by $- 1$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 \sqrt{2 x} - 4 \sqrt{2})}$

Step 8

Factor $2$ out of $2 \sqrt{2 x} - 4 \sqrt{2}$ .

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Step 8.1

Factor $2$ out of $2 \sqrt{2 x}$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 (\sqrt{2 x}) - 4 \sqrt{2})}$

Step 8.2

Factor $2$ out of $- 4 \sqrt{2}$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 (\sqrt{2 x}) + 2 (- 2 \sqrt{2}))}$

Step 8.3

Factor $2$ out of $2 (\sqrt{2 x}) + 2 (- 2 \sqrt{2})$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 (2 (\sqrt{2 x} - 2 \sqrt{2}))}$

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2 \cdot (2 (\sqrt{2 x} - 2 \sqrt{2}))}$

Step 9

Write the expression using exponents.

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Step 9.1

Multiply $2$ by $2$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{4 (\sqrt{2 x} - 2 \sqrt{2})}$

Step 9.2

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{8 x} - \sqrt{32}) = \sqrt{2^{2} (\sqrt{2 x} - 2 \sqrt{2})}$

Step 10

Pull terms out from under the radical.

$f (\sqrt{8 x} - \sqrt{32}) = 2 \sqrt{\sqrt{2 x} - 2 \sqrt{2}}$