https://file.aiquickdraw.com/mathbot/file/1727423379535tsvedrlz.jpg

It seems there was an issue opening the image. Let me try again to extract the information.It appears there is an issue with opening the image. Let me directly describe and analyze the contents based on the text I can interpret:

The question asks to find the domain and range for three functions:
1. \( f(x) = \sqrt{2x + 3} \)
2. \( g(x) = \frac{1}{4x - 1} \)
3. \( h(x) = \sqrt{(x + 1)^{-1}} \)

### Solutions:

1. **For \( f(x) = \sqrt{2x + 3} \):**
   - **Domain**: Since square roots require the argument to be non-negative, we need \( 2x + 3 \geq 0 \). Solving this gives \( x \geq -\frac{3}{2} \). Therefore, the domain is:
     \[
     x \in \left[ -\frac{3}{2}, \infty \right)
     \]
   - **Range**: The output of a square root function is non-negative, so the range is:
     \[
     f(x) \geq 0 \quad \text{or} \quad f(x) \in [0, \infty)
     \]

2. **For \( g(x) = \frac{1}{4x - 1} \):**
   - **Domain**: The function is undefined when the denominator is zero, i.e., \( 4x - 1 = 0 \) implies \( x = \frac{1}{4} \). Thus, the domain is all real numbers except \( x = \frac{1}{4} \):
     \[
     x \in (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)
     \]
   - **Range**: Since this is a rational function, \( g(x) \) can take any real value except 0, which is asymptotically approached but never reached. Hence, the range is:
     \[
     g(x) \in (-\infty, 0) \cup (0, \infty)
     \]

3. **For \( h(x) = \sqrt{(x + 1)^{-1}} \):**
   - **Domain**: The argument of the square root must be non-negative, i.e., \( (x + 1)^{-1} \geq 0 \). This implies that \( x + 1 > 0 \), so \( x > -1 \). Therefore, the domain is:
     \[
     x \in (-1, \infty)
     \]
   - **Range**: Since the square root function is non-negative, the range of this function is:
     \[
     h(x) \in [0, \infty)
     \]

Would you like more details or clarifications? Here are five related questions to further explore these concepts:

1. How do we determine the domain of a function involving a square root?
2. What does the behavior of rational functions near vertical asymptotes tell us about their range?
3. How would the domain of a function change if the expression under the square root was negative?
4. Can you explain why \( g(x) \) never equals zero?
5. How do the properties of inverse powers, like in \( h(x) = \sqrt{(x+1)^{-1}} \), influence the function's domain?

**Tip**: When finding the domain of functions involving square roots, always start by solving the inequality inside the square root for non-negative values.

Tentukan domain dan range dari: a. f(x) = √(2x + 3), b. g(x) = 1 / (4x - 1), c. h(x) = √((x + 1)^-1)

Explore how to determine the domain and range of three functions: f(x) = √(2x + 3), g(x) = 1 / (4x - 1), and h(x) = √((x + 1)^-1). Understand the constraints on square root and rational functions, and learn to solve inequalities to find the domain.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation