https://file.aiquickdraw.com/mathbot/file/1726813920134mg7dghfz.jpg

Let's address each of the two problems in the image.

### Problem 1:
The function \( h(x) = (3 + x)^2 \) has its domain as \( [-3, 3] \).

#### Does \( h \) have an inverse?
To determine if \( h \) has an inverse, we need to check if it's a one-to-one function. A function is one-to-one if it passes the **horizontal line test**, meaning that no horizontal line intersects the graph of the function at more than one point.

For \( h(x) = (3 + x)^2 \), the function is a **quadratic** (a parabola) that opens upwards, which is not one-to-one over the interval \( [-3, 3] \). This is because for any \( y \)-value within the range, there are generally two different \( x \)-values that can produce the same output \( h(x) \).

**Domain and Range**:
- **Domain**: \( [-3, 3] \)
- **Range**: Since the function \( h(x) = (3 + x)^2 \) achieves its minimum value at \( x = -3 \) and maximum value at \( x = 3 \), the range of \( h(x) \) is \( [0, 36] \).

Thus, since the function is not one-to-one, it does **not** have an inverse.

### Problem 2:
The function \( g(x) = (5x^2 + 3)^{7777} \) has a domain of \( [-2, 2] \).

#### Explain why \( g \) is not a one-to-one function:
The expression inside the parentheses is \( 5x^2 + 3 \), which is even and symmetric about \( x = 0 \). Since \( x^2 \) is the same for \( x \) and \( -x \), the function \( g(x) \) will output the same value for both positive and negative values of \( x \). This symmetry shows that \( g(x) \) is **not one-to-one** because different \( x \)-values lead to the same output.

Additionally, raising \( (5x^2 + 3) \) to the large exponent 7777 does not affect the fact that the function is even. Therefore, \( g(x) \) fails the horizontal line test and is not one-to-one.

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Do you have any specific questions or need further clarifications?

Here are 5 related questions to explore:

1. What is the general condition for a function to have an inverse?
2. How does the range of a quadratic function influence whether it has an inverse?
3. Why does the symmetry in \( g(x) = (5x^2 + 3)^{7777} \) ensure it is not one-to-one?
4. How would the behavior of \( g(x) \) change if the exponent were odd instead of even?
5. Can you give an example of a quadratic function that is one-to-one over a restricted domain?

**Tip**: The horizontal line test is a quick way to visually determine if a function is one-to-one and, therefore, has an inverse.

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. Suppose g is the function whose domain is the interval [-2, 2], with g defined on this domain by the formula g(x) = (5x^2 + 3)^7777. Explain why g is not a one-to-one function.

Explore whether h(x) = (3 + x)^2 has an inverse and understand why g(x) = (5x^2 + 3)^7777 is not a one-to-one function. This problem delves into inverse functions, the horizontal line test, and quadratic functions. Suitable for students in Grades 10-12.

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