Calculus Examples

$f (x) = \sqrt{3 x - 5}$ , $h (x) = \frac{x}{1 + x^{2}}$ , $(f \circ h)$

Step 1

Set up the composite result function.

$f (h (x))$

Step 2

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

$f (\frac{x}{1 + x^{2}}) = \sqrt{3 (\frac{x}{1 + x^{2}}) - 5}$

Step 3

Combine $3$ and $\frac{x}{1 + x^{2}}$ .

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x}{1 + x^{2}} - 5}$

Step 4

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x}{1 + x^{2}} - 5 \cdot \frac{1 + x^{2}}{1 + x^{2}}}$

Step 5

Combine $- 5$ and $\frac{1 + x^{2}}{1 + x^{2}}$ .

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x}{1 + x^{2}} + \frac{- 5 (1 + x^{2})}{1 + x^{2}}}$

Step 6

Combine the numerators over the common denominator.

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x - 5 (1 + x^{2})}{1 + x^{2}}}$

Step 7

Reorder terms.

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x - 5 (1 + x^{2})}{x^{2} + 1}}$

Step 8

Rewrite $\frac{3 x - 5 (1 + x^{2})}{x^{2} + 1}$ in a factored form.

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Step 8.1

Apply the distributive property.

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x - 5 \cdot 1 - 5 x^{2}}{x^{2} + 1}}$

Step 8.2

Multiply $- 5$ by $1$ .

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{3 x - 5 - 5 x^{2}}{x^{2} + 1}}$

Step 8.3

Reorder terms.

$f (\frac{x}{1 + x^{2}}) = \sqrt{\frac{- 5 x^{2} + 3 x - 5}{x^{2} + 1}}$

Step 9

Rewrite $\sqrt{\frac{- 5 x^{2} + 3 x - 5}{x^{2} + 1}}$ as $\frac{\sqrt{- 5 x^{2} + 3 x - 5}}{\sqrt{x^{2} + 1}}$ .

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5}}{\sqrt{x^{2} + 1}}$

Step 10

Multiply $\frac{\sqrt{- 5 x^{2} + 3 x - 5}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5}}{\sqrt{x^{2} + 1}} \cdot \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{\sqrt{- 5 x^{2} + 3 x - 5}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 11.2

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 11.3

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 11.4

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{2}}$

Step 11.6

Rewrite ${\sqrt{x^{2} + 1}}^{2}$ as $x^{2} + 1$ .

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Step 11.6.1

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{({(x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 11.6.2

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 11.6.3

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

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 11.6.4.2

Rewrite the expression.

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 11.6.5

Simplify.

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{- 5 x^{2} + 3 x - 5} \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12

Combine using the product rule for radicals.

$f (\frac{x}{1 + x^{2}}) = \frac{\sqrt{(- 5 x^{2} + 3 x - 5) (x^{2} + 1)}}{x^{2} + 1}$