Algebra Examples

$f (x) = \sqrt{4 x - 1}$ , $g (x) = \frac{1}{x}$

Step 1

Find the sum of the functions.

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Step 1.1

Replace the function designators with the actual functions in $f (x) + g (x)$ .

$\sqrt{4 x - 1} + \frac{1}{x}$

Step 1.2

Simplify.

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Step 1.2.1

To write $\sqrt{4 x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\sqrt{4 x - 1} x}{x} + \frac{1}{x}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{\sqrt{4 x - 1} x + 1}{x}$

Step 2

Set the radicand in $\sqrt{4 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$4 x - 1 \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

Add $1$ to both sides of the inequality.

$4 x \geq 1$

Step 3.2

Divide each term in $4 x \geq 1$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x \geq 1$ by $4$ .

$\frac{4 x}{4} \geq \frac{1}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{1}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{4}$

Step 4

Set the denominator in $\frac{\sqrt{4 x - 1} x + 1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5

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

Interval Notation:

$[\frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{1}{4}}$

Step 6