Find f(g(x)) f(x) = square root of 2/(x-3) g(x)=x^2-7

$f (x) = \sqrt{\frac{2}{x - 3}}$ $g (x) = x^{2} - 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} - 7)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} - 7) = \sqrt{\frac{2}{(x^{2} - 7) - 3}}$

Step 3

Subtract $3$ from $- 7$ .

$f (x^{2} - 7) = \sqrt{\frac{2}{x^{2} - 10}}$

Step 4

Rewrite $\sqrt{\frac{2}{x^{2} - 10}}$ as $\frac{\sqrt{2}}{\sqrt{x^{2} - 10}}$ .

$f (x^{2} - 7) = \frac{\sqrt{2}}{\sqrt{x^{2} - 10}}$

Step 5

Multiply $\frac{\sqrt{2}}{\sqrt{x^{2} - 10}}$ by $\frac{\sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10}}$ .

$f (x^{2} - 7) = \frac{\sqrt{2}}{\sqrt{x^{2} - 10}} \cdot \frac{\sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{x^{2} - 10}}$ by $\frac{\sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10}}$ .

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10} \sqrt{x^{2} - 10}}$

Raise $\sqrt{x^{2} - 10}$ to the power of $1$ .

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10} \sqrt{x^{2} - 10}}$

Raise $\sqrt{x^{2} - 10}$ to the power of $1$ .

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{\sqrt{x^{2} - 10} \sqrt{x^{2} - 10}}$

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

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{\sqrt{x^{2} - 10}}^{1 + 1}}$

Add $1$ and $1$ .

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{\sqrt{x^{2} - 10}}^{2}}$

Rewrite ${\sqrt{x^{2} - 10}}^{2}$ as $x^{2} - 10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 10}$ as ${(x^{2} - 10)}^{\frac{1}{2}}$ .

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{({(x^{2} - 10)}^{\frac{1}{2}})}^{2}}$

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

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{(x^{2} - 10)}^{\frac{1}{2} \cdot 2}}$

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

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{(x^{2} - 10)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{{(x^{2} - 10)}^{\frac{2}{2}}}$

Rewrite the expression.

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{x^{2} - 10}$

Simplify.

$f (x^{2} - 7) = \frac{\sqrt{2} \sqrt{x^{2} - 10}}{x^{2} - 10}$

Step 7

Combine using the product rule for radicals.

$f (x^{2} - 7) = \frac{\sqrt{2 (x^{2} - 10)}}{x^{2} - 10}$