Find the inverse of each relation and the domain and range. Y=√ x + 1 over 3

To find the inverse of the given relation \( y = \frac{\sqrt{x} + 1}{3} \), let's go through the steps and then determine the domain and range.

### Step 1: Rewrite the equation
We start with the equation:
\[
y = \frac{\sqrt{x} + 1}{3}
\]

### Step 2: Solve for \( x \) in terms of \( y \)

Multiply both sides by 3 to eliminate the fraction:
\[
3y = \sqrt{x} + 1
\]

Subtract 1 from both sides:
\[
3y - 1 = \sqrt{x}
\]

Now square both sides to remove the square root:
\[
(3y - 1)^2 = x
\]

So, the inverse of the function is:
\[
x = (3y - 1)^2
\]
Or, equivalently, the inverse relation is:
\[
y^{-1}(x) = (3x - 1)^2
\]

### Step 3: Determine the domain and range

- For the original function \( y = \frac{\sqrt{x} + 1}{3} \), the square root term \( \sqrt{x} \) requires \( x \geq 0 \). Therefore, the **domain** of the original function is \( [0, \infty) \).

  The square root ensures that the output is non-negative, and dividing by 3 does not affect the positivity, so the **range** of the original function is \( [\frac{1}{3}, \infty) \).

- For the inverse function \( y^{-1}(x) = (3x - 1)^2 \), we must ensure that the expression inside the square is non-negative. The **domain** of the inverse function is \( [\frac{1}{3}, \infty) \), and since squaring any real number gives a non-negative result, the **range** of the inverse is \( [0, \infty) \).

### Summary:
- The inverse relation is: \( y^{-1}(x) = (3x - 1)^2 \)
- Domain of the original function: \( [0, \infty) \)
- Range of the original function: \( \left[ \frac{1}{3}, \infty \right) \)
- Domain of the inverse function: \( \left[ \frac{1}{3}, \infty \right) \)
- Range of the inverse function: \( [0, \infty) \)

Do you have any questions or want more details?

### Related Questions:
1. How do you find the inverse of a logarithmic function?
2. What are the conditions for a function to have an inverse?
3. Can all square root functions have inverses? Why or why not?
4. How does the domain of a function affect its inverse?
5. How can you determine if a relation is a function?

**Tip:** Always ensure that the function has a one-to-one correspondence before attempting to find its inverse.

Find the inverse of each relation and the domain and range. Y = √x + 1 over 3

Discover how to find the inverse of the function y = (√x + 1)/3 and determine its domain and range. This step-by-step guide covers the inverse function, domain, range, and properties of square root functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation