https://file.aiquickdraw.com/mathbot/file/1726873133417yebdzym3.jpg

The question asks whether the given relation defines \( y \) as a function of \( x \), and also asks for the domain and range of the relation.

The given equation is:
\[
y = \sqrt{8x + 3}
\]

### Step 1: Is it a function?
A relation is a function if for every input \( x \), there is exactly one output \( y \). 

For this equation, the square root function \( \sqrt{} \) always returns only the non-negative square root (the principal root). Therefore, for every valid \( x \), there will be only one value of \( y \), making this relation a function. Thus, the answer to the question "Does the relation define a function?" is **Yes**.

### Step 2: Domain
The domain of the function refers to the set of all possible \( x \)-values that make the expression inside the square root non-negative. In this case, we need:
\[
8x + 3 \geq 0
\]

Solving for \( x \):
\[
8x \geq -3 \quad \Rightarrow \quad x \geq -\frac{3}{8}
\]

Thus, the domain in interval notation is:
\[
\left[ -\frac{3}{8}, \infty \right)
\]

### Step 3: Range
Since the square root function only gives non-negative outputs, the smallest value for \( y \) is 0 (which happens when \( x = -\frac{3}{8} \)). The range is therefore all non-negative numbers:
\[
[0, \infty)
\]

### Final Answers:
1. **Does the relation define a function?** Yes.
2. **Domain:** \( \left[ -\frac{3}{8}, \infty \right) \)
3. **Range:** \( [0, \infty) \)

Would you like further details or have any other questions?

### Related Questions:
1. How do you find the domain of more complex functions?
2. What happens if the equation includes a denominator or logarithm?
3. Can a square root function ever have negative values in its range?
4. How can we graph \( y = \sqrt{8x + 3} \) to confirm its domain and range?
5. How does the concept of a function apply to real-world examples?

### Tip:
Always check for restrictions such as square roots, denominators, and logarithms when determining the domain of a function.

Decide whether the given relation defines y as a function of x. Give the domain and range.
y = √(8x + 3)
Does the relation define a function?

Explore the function y = √(8x + 3), determine if it defines a valid function, and learn how to calculate its domain and range. Ideal for students in Grades 9-11, this problem delves into square roots, domain restrictions, and the concept of a function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation