Find g(f(x)) f(x) = square root of x g(x)=1/( square root of x) h(x)=x^2

$f (x) = \sqrt{x}$ $g (x) = \frac{1}{\sqrt{x}}$ $h (x) = x^{2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

$g (\sqrt{x}) = \frac{1}{\sqrt{\sqrt{x}}}$

Step 3

Remove parentheses.

$g (\sqrt{x}) = \frac{1}{\sqrt{\sqrt{x}}}$

Step 4

Rewrite $\sqrt{\sqrt{x}}$ as $\sqrt[4]{x}$ .

$g (\sqrt{x}) = \frac{1}{\sqrt[4]{x}}$

Step 5

Multiply $\frac{1}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$g (\sqrt{x}) = \frac{1}{\sqrt[4]{x}} \cdot \frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{\sqrt[4]{x} {\sqrt[4]{x}}^{3}}$

Raise $\sqrt[4]{x}$ to the power of $1$ .

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{\sqrt[4]{x} {\sqrt[4]{x}}^{3}}$

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

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{1 + 3}}$

Add $1$ and $3$ .

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{4}}$

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{{(x^{\frac{1}{4}})}^{4}}$

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

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{x^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Rewrite the expression.

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{x}$

Simplify.

$g (\sqrt{x}) = \frac{{\sqrt[4]{x}}^{3}}{x}$

Step 7

Rewrite ${\sqrt[4]{x}}^{3}$ as $\sqrt[4]{x^{3}}$ .

$g (\sqrt{x}) = \frac{\sqrt[4]{x^{3}}}{x}$