Find g(f(x)) f(x)=1/(x^2+23) , g(x) = square root of x+100

$f (x) = \frac{1}{x^{2} + 23}$ , $g (x) = \sqrt{x + 100}$

Set up the composite result function.

$g (f (x))$

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

$g (\frac{1}{x^{2} + 23}) = \sqrt{(\frac{1}{x^{2} + 23}) + 100}$

To write $100$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 23}{x^{2} + 23}$ .

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{1}{x^{2} + 23} + \frac{100 (x^{2} + 23)}{x^{2} + 23}}$

Combine the numerators over the common denominator.

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{1 + 100 (x^{2} + 23)}{x^{2} + 23}}$

Reorder terms.

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{100 (x^{2} + 23) + 1}{x^{2} + 23}}$

Rewrite $\frac{100 (x^{2} + 23) + 1}{x^{2} + 23}$ in a factored form.

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Apply the distributive property.

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{100 x^{2} + 100 \cdot 23 + 1}{x^{2} + 23}}$

Multiply $100$ by $23$ .

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{100 x^{2} + 2300 + 1}{x^{2} + 23}}$

Add $2300$ and $1$ .

$g (\frac{1}{x^{2} + 23}) = \sqrt{\frac{100 x^{2} + 2301}{x^{2} + 23}}$

Rewrite $\sqrt{\frac{100 x^{2} + 2301}{x^{2} + 23}}$ as $\frac{\sqrt{100 x^{2} + 2301}}{\sqrt{x^{2} + 23}}$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301}}{\sqrt{x^{2} + 23}}$

Multiply $\frac{\sqrt{100 x^{2} + 2301}}{\sqrt{x^{2} + 23}}$ by $\frac{\sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23}}$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301}}{\sqrt{x^{2} + 23}} \cdot \frac{\sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23}}$

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{100 x^{2} + 2301}}{\sqrt{x^{2} + 23}}$ and $\frac{\sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23}}$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23} \sqrt{x^{2} + 23}}$

Raise $\sqrt{x^{2} + 23}$ to the power of $1$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23} \sqrt{x^{2} + 23}}$

Raise $\sqrt{x^{2} + 23}$ to the power of $1$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{\sqrt{x^{2} + 23} \sqrt{x^{2} + 23}}$

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

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{\sqrt{x^{2} + 23}}^{1 + 1}}$

Add $1$ and $1$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{\sqrt{x^{2} + 23}}^{2}}$

Rewrite ${\sqrt{x^{2} + 23}}^{2}$ as $x^{2} + 23$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 23}$ as ${(x^{2} + 23)}^{\frac{1}{2}}$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{({(x^{2} + 23)}^{\frac{1}{2}})}^{2}}$

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

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{(x^{2} + 23)}^{\frac{1}{2} \cdot 2}}$

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

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{(x^{2} + 23)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{{(x^{2} + 23)}^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{x^{2} + 23}$

Simplify.

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{100 x^{2} + 2301} \sqrt{x^{2} + 23}}{x^{2} + 23}$

Combine using the product rule for radicals.

$g (\frac{1}{x^{2} + 23}) = \frac{\sqrt{(100 x^{2} + 2301) (x^{2} + 23)}}{x^{2} + 23}$

Find g(f(x)) f(x)=1/(x^2+23) , g(x) = square root of x+100

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