domain of f(x)= square root of 8-x, g(x)=1/x-3

To find the domains of the functions:

### 1. Domain of \( f(x) = \sqrt{8 - x} \)
- The square root function is only defined for non-negative values inside the square root. Therefore, we need:
  \[
  8 - x \geq 0
  \]
  Solving this inequality:
  \[
  x \leq 8
  \]
  So, the domain of \( f(x) \) is:
  \[
  (-\infty, 8]
  \]

### 2. Domain of \( g(x) = \frac{1}{x - 3} \)
- The function \( g(x) \) is a rational function, and it is undefined when the denominator equals zero. So, we need to exclude the value where \( x - 3 = 0 \), which is \( x = 3 \).
  The domain of \( g(x) \) is all real numbers except 3:
  \[
  (-\infty, 3) \cup (3, \infty)
  \]

### Final Results:
- **Domain of \( f(x) \)**: \( (-\infty, 8] \)
- **Domain of \( g(x) \)**: \( (-\infty, 3) \cup (3, \infty) \)

Would you like any details or have any further questions?

Here are 5 related questions to explore:
1. What is the range of \( f(x) = \sqrt{8 - x} \)?
2. Can you find the composition of \( f(g(x)) \)?
3. How do we determine the domain of a product of two functions?
4. What happens to the graph of \( f(x) = \sqrt{8 - x} \) as \( x \) approaches 8?
5. What is the vertical asymptote of \( g(x) = \frac{1}{x - 3} \)?

**Tip:** Always check for restrictions like square roots or denominators when finding a function's domain.

Learn how to find the domain of f(x) = sqrt(8 - x) and g(x) = 1 / (x - 3) with a detailed step-by-step solution. This problem explores function domains, including square root and rational functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation