Find the Domain of the Quotient of the Two Functions f(x)=2x-5 , g(x) = square root of x+3

$f (x) = 2 x - 5$ , $g (x) = \sqrt{x + 3}$

Step 1

Find the quotient of the functions.

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

$\frac{2 x - 5}{\sqrt{x + 3}}$

Simplify.

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

$\frac{2 x - 5}{\sqrt{x + 3}} \cdot \frac{\sqrt{x + 3}}{\sqrt{x + 3}}$

Combine and simplify the denominator.

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

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

Raise $\sqrt{x + 3}$ to the power of $1$ .

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

Raise $\sqrt{x + 3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Reorder factors in $\frac{(2 x - 5) \sqrt{x + 3}}{x + 3}$ .

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

Step 2

Set the radicand in $\sqrt{x + 3}$ greater than or equal to $0$ to find where the expression is defined.

$x + 3 \geq 0$

Step 3

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

Step 4

Set the denominator in $\frac{\sqrt{x + 3} (2 x - 5)}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 5

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

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

Interval Notation:

$(- 3, \infty)$

Set-Builder Notation:

${x | x > - 3}$

Step 7

Find the Domain of the Quotient of the Two Functions f(x)=2x-5 , g(x) = square root of x+3

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