https://file.aiquickdraw.com/mathbot/file/1725946340604u303o18d.jpg

To find the domain of the function

\[
f(x) = \frac{\sqrt{5 + x}}{6 - 2x}
\]

we need to determine when both the square root and the denominator are defined.

### 1. Condition from the square root:
The expression inside the square root must be non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore:

\[
5 + x \geq 0
\]

Solving for \( x \):

\[
x \geq -5
\]

### 2. Condition from the denominator:
The denominator must not be zero, since division by zero is undefined. Therefore:

\[
6 - 2x \neq 0
\]

Solving for \( x \):

\[
x \neq 3
\]

### Conclusion:
The domain of \( f(x) \) is the set of values of \( x \) where both conditions are satisfied. Combining the results:

\[
x \geq -5 \quad \text{and} \quad x \neq 3
\]

This gives us the domain:

\[
[-5, 3) \cup (3, \infty)
\]

Thus, the domain is the union of the intervals \([-5, 3)\) and \((3, \infty)\).

Would you like further clarification or have any other questions?

### Related Questions:
1. How would the domain change if the square root were replaced with an absolute value?
2. What happens if the denominator also involves a square root term?
3. Can you express this function in a simplified form without altering the domain?
4. What is the range of this function given its domain?
5. How would the function behave near \(x = 3\), where the denominator approaches zero?

### Tip:
When determining the domain of a function, always check for restrictions caused by square roots and denominators.

Find the domain of the function: f(x) = (√(5 + x))/(6 - 2x).

Learn how to determine the domain of the function f(x) = (√(5 + x))/(6 - 2x). This problem explores key mathematical concepts such as functions, square roots, and inequalities, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation