https://file.aiquickdraw.com/mathbot/file/1727113233094x3gx43nv.jpg

The problem presented is as follows:

1. **Consider the function \( h \) whose domain is the interval \([-3, 3]\), with \( h \) defined on this domain by the formula:**
   \[
   h(x) = (3 + x)^2
   \]

   The question asks: **Does \( h \) have an inverse? If so, find it, along with its domain and range. If not, explain why not.**

### Analysis:

To determine if the function \( h(x) = (3 + x)^2 \) has an inverse, we need to check if the function is **one-to-one** on the domain \([-3, 3]\). A function has an inverse if and only if it is **injective**, which means that different values of \( x \) must map to different values of \( h(x) \). 

1. **Injectivity Test:**
   The function \( h(x) = (3 + x)^2 \) is a quadratic function, which is parabolic and symmetric about \( x = -3 \). This means that for two values \( x_1 \) and \( x_2 \), it is possible for \( h(x_1) = h(x_2) \) even if \( x_1 \neq x_2 \). Thus, the function is **not injective**.

2. **Checking on the domain \([-3, 3]\):**
   - For example, at \( x = -3 \), we have \( h(-3) = (3 - 3)^2 = 0 \).
   - At \( x = 3 \), we get \( h(3) = (3 + 3)^2 = 36 \).
   - In between, the values repeat due to the symmetry of the parabola. Therefore, \( h \) is not one-to-one over the domain \([-3, 3]\).

Thus, the function \( h(x) \) **does not have an inverse** over the entire domain \([-3, 3]\).

---

Would you like further explanation on the concept of inverse functions or details about this problem?

Here are some related questions that can expand on this topic:

1. What is the graphical interpretation of a function that has an inverse?
2. How can we restrict the domain of \( h(x) \) to make it one-to-one?
3. What are the general conditions for a quadratic function to have an inverse?
4. How do we algebraically find the inverse of a function when it exists?
5. What are the differences between injective, surjective, and bijective functions?

**Tip:** To determine if a function is invertible, remember that a horizontal line should intersect the graph at most once (this is the horizontal line test).

Consider the function h whose domain is the interval [-3, 3], with h defined on this domain by the formula h(x) = (3 + x)^2. Does h have an inverse? If so, find it, along with its domain and range. If not, explain why not.

Explore whether the function h(x) = (3 + x)^2 defined on the interval [-3, 3] has an inverse. This problem tests knowledge of quadratic equations and inverse functions, focusing on injectivity and the horizontal line test. Suitable for 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