Finite Math Examples

$g (x) = \frac{3 x - 5}{x + 1}$ , $h (y) = \sqrt{1 - 3 y}$ , $h (g (0))$

Step 1

Set up the composite result function.

$h (g (x))$

Step 2

Evaluate $h (\frac{3 x - 5}{x + 1})$ by substituting in the value of $g$ into $h$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{1 - 3 \frac{3 x - 5}{x + 1}}$

Step 3

Combine $- 3$ and $\frac{3 x - 5}{x + 1}$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{1 + \frac{- 3 (3 x - 5)}{x + 1}}$

Step 4

Simplify the expression.

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Step 4.1

Move the negative in front of the fraction.

$h (\frac{3 x - 5}{x + 1}) = \sqrt{1 - \frac{3 (3 x - 5)}{x + 1}}$

Step 4.2

Write $1$ as a fraction with a common denominator.

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{x + 1}{x + 1} - \frac{3 (3 x - 5)}{x + 1}}$

Step 5

Combine the numerators over the common denominator.

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{x + 1 - 3 (3 x - 5)}{x + 1}}$

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 3$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{x + 1 - 9 x - 3 \cdot - 5}{x + 1}}$

Step 6.3

Multiply $- 3$ by $- 5$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{x + 1 - 9 x + 15}{x + 1}}$

Step 6.4

Subtract $9 x$ from $x$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{- 8 x + 1 + 15}{x + 1}}$

Step 6.5

Add $1$ and $15$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{- 8 x + 16}{x + 1}}$

Step 6.6

Factor $8$ out of $- 8 x + 16$ .

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Step 6.6.1

Factor $8$ out of $- 8 x$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{8 (- x) + 16}{x + 1}}$

Step 6.6.2

Factor $8$ out of $16$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{8 (- x) + 8 (2)}{x + 1}}$

Step 6.6.3

Factor $8$ out of $8 (- x) + 8 (2)$ .

$h (\frac{3 x - 5}{x + 1}) = \sqrt{\frac{8 (- x + 2)}{x + 1}}$

Step 7

Rewrite $\sqrt{\frac{8 (- x + 2)}{x + 1}}$ as $\frac{\sqrt{8 (- x + 2)}}{\sqrt{x + 1}}$ .

$h (\frac{3 x - 5}{x + 1}) = \frac{\sqrt{8 (- x + 2)}}{\sqrt{x + 1}}$

Step 8

Simplify the numerator.

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Step 8.1

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

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Step 8.1.1

Factor $4$ out of $8$ .

$h (\frac{3 x - 5}{x + 1}) = \frac{\sqrt{4 (2) (- x + 2)}}{\sqrt{x + 1}}$

Step 8.1.2

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

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

Step 8.1.3

Add parentheses.

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

Step 8.2

Pull terms out from under the radical.

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

Step 9

Multiply $\frac{2 \sqrt{2 (- x + 2)}}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

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

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{2 \sqrt{2 (- x + 2)}}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Add $1$ and $1$ .

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

Step 10.6

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

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Step 10.6.1

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

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

Step 10.6.2

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

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

Step 10.6.3

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

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

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

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

Step 10.6.4.2

Rewrite the expression.

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

Step 10.6.5

Simplify.

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

Step 11

Combine using the product rule for radicals.

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

Step 12

Move $2$ to the left of $x + 1$ .

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

Step 13

Evaluate the result function by replacing $x$ with $0$ in the function.

$\frac{2 \sqrt{2 ((0) + 1) (- (0) + 2)}}{(0) + 1}$

Step 14

Simplify the evaluated function.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $- 1$ by $0$ .

$\frac{2 \sqrt{2 (0 + 1) (0 + 2)}}{0 + 1}$

Step 14.1.2

Add $0$ and $1$ .

$\frac{2 \sqrt{2 \cdot 1 (0 + 2)}}{0 + 1}$

Step 14.1.3

Multiply $2$ by $1$ .

$\frac{2 \sqrt{2 (0 + 2)}}{0 + 1}$

Step 14.1.4

Add $0$ and $2$ .

$\frac{2 \sqrt{2 \cdot 2}}{0 + 1}$

Step 14.1.5

Multiply $2$ by $2$ .

$\frac{2 \sqrt{4}}{0 + 1}$

Step 14.1.6

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

$\frac{2 \sqrt{2^{2}}}{0 + 1}$

Step 14.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 \cdot 2}{0 + 1}$

Step 14.2

Simplify the expression.

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Step 14.2.1

Add $0$ and $1$ .

$\frac{2 \cdot 2}{1}$

Step 14.2.2

Multiply $2$ by $2$ .

$\frac{4}{1}$

Step 14.2.3

Divide $4$ by $1$ .

$4$