Find the Domain of the Quotient of the Two Functions f(x)=8x+1 , g(x) = square root of x

$f (x) = 8 x + 1$ , $g (x) = \sqrt{x}$

Step 1

Find the quotient of the functions.

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{8 x + 1}{\sqrt{x}}$

Simplify.

Multiply $\frac{8 x + 1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{8 x + 1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Combine and simplify the denominator.

Multiply $\frac{8 x + 1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{(8 x + 1) \sqrt{x}}{\sqrt{x} \sqrt{x}}$

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

$\frac{(8 x + 1) \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

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

$\frac{(8 x + 1) \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

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

$\frac{(8 x + 1) \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(8 x + 1) \sqrt{x}}{{\sqrt{x}}^{2}}$

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

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

$\frac{(8 x + 1) \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

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

$\frac{(8 x + 1) \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

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

$\frac{(8 x + 1) \sqrt{x}}{x^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(8 x + 1) \sqrt{x}}{x^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(8 x + 1) \sqrt{x}}{x^{1}}$

Simplify.

$\frac{(8 x + 1) \sqrt{x}}{x}$

Reorder factors in $\frac{(8 x + 1) \sqrt{x}}{x}$ .

$\frac{\sqrt{x} (8 x + 1)}{x}$

Step 2

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 3

Set the denominator in $\frac{\sqrt{x} (8 x + 1)}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 5

Find the Domain of the Quotient of the Two Functions f(x)=8x+1 , g(x) = square root of x

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