Question
find the domain of the following function: f(x) = √x+3/ x-6.
[-3,6) U (6,oo)
[-3,oo)
x ≠ 6
(6,oo)
All real numbers
[-3,6) U (6,oo)
[-3,oo)
x ≠ 6
(6,oo)
All real numbers
Answers
GPT-4o mini
To find the domain of the function \( f(x) = \frac{\sqrt{x + 3}}{x - 6} \), we need to consider the conditions under which the function is defined.
1. **Square Root Condition**: The expression inside the square root must be non-negative:
\[
x + 3 \geq 0 \implies x \geq -3
\]
2. **Denominator Condition**: The denominator cannot be zero, so we must avoid the value that makes it zero:
\[
x - 6 \neq 0 \implies x \neq 6
\]
Combining these conditions, we find:
- The square root condition gives \( x \geq -3 \).
- The denominator condition excludes \( x = 6 \).
So the potential values of \( x \) which satisfy the first condition are \([-3, \infty)\), but we exclude \( x = 6\).
Thus, the domain of the function is:
\[
[-3, 6) \cup (6, \infty)
\]
Therefore, the correct answer is:
\[
[-3,6) \cup (6,\infty)
\]
1. **Square Root Condition**: The expression inside the square root must be non-negative:
\[
x + 3 \geq 0 \implies x \geq -3
\]
2. **Denominator Condition**: The denominator cannot be zero, so we must avoid the value that makes it zero:
\[
x - 6 \neq 0 \implies x \neq 6
\]
Combining these conditions, we find:
- The square root condition gives \( x \geq -3 \).
- The denominator condition excludes \( x = 6 \).
So the potential values of \( x \) which satisfy the first condition are \([-3, \infty)\), but we exclude \( x = 6\).
Thus, the domain of the function is:
\[
[-3, 6) \cup (6, \infty)
\]
Therefore, the correct answer is:
\[
[-3,6) \cup (6,\infty)
\]