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.