Questions: What is the domain of the following function: f(x)=sqrt(x+8)/(x-2). [-8,2) ∪(2, ∞) [-8, ∞) (2, ∞) x ≠ 2 All real numbers

Transcript text: What is the domain of the following function: $f(x)=\frac{\sqrt{x+8}}{x-2}$. $[-8,2) \cup(2, \infty)$ $[-8, \infty)$ $(2, \infty)$ $x \neq 2$ All real numbers

Solution

Solution Steps

Step 1: Identify the restrictions on the domain

The function $ f(x) = \frac{\sqrt{x+8}}{x-2} $ has two restrictions:

The expression under the square root must be non-negative: $ x + 8 \geq 0 $.
The denominator cannot be zero: $ x - 2 \neq 0 $.

Step 2: Solve the inequality for the square root

To ensure the expression under the square root is non-negative: \[ x + 8 \geq 0 \implies x \geq -8. \] This means $ x $ must be greater than or equal to $-8$.

Step 3: Solve the inequality for the denominator

To ensure the denominator is not zero: \[ x - 2 \neq 0 \implies x \neq 2. \] This means $ x $ cannot be equal to $2$.

Step 4: Combine the restrictions

Combining the two restrictions, the domain of $ f(x) $ is all real numbers $ x $ such that $ x \geq -8 $ and $ x \neq 2 $. In interval notation, this is: \[ [-8, 2) \cup (2, \infty). \]