Algebra Examples

$f (x) = \sqrt{\frac{1}{2} x - 10} + 3$

Step 1

Set the radicand in $\sqrt{\frac{1}{2} x - 10}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{1}{2} x - 10 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$\frac{x}{2} - 10 \geq 0$

Step 2.2

Add $10$ to both sides of the inequality.

$\frac{x}{2} \geq 10$

Step 2.3

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq 10 \cdot 2$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq 10 \cdot 2$

Step 2.4.1.1.2

Rewrite the expression.

$x \geq 10 \cdot 2$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $10$ by $2$ .

$x \geq 20$

Step 3

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

Interval Notation:

$[20, \infty)$

Set-Builder Notation:

${x | x \geq 20}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${y | y \geq 3}$

Step 5

Determine the domain and range.

Domain: $[20, \infty), {x | x \geq 20}$

Range: $[3, \infty), {y | y \geq 3}$

Step 6