https://file.aiquickdraw.com/mathbot/file/17261372695197lzqpdp4.jpg

Let's go through the two problems shown in the image step-by-step.

### Problem 1: Function \( f(x) = \frac{x - 2}{x - 3} \)
You are asked to find \( c \) such that \( f(c) = 2 \) in the interval \([-3, 4]\).

The function is given by:
\[
f(x) = \frac{x - 2}{x - 3}
\]
We need to find \( c \) such that:
\[
f(c) = 2
\]
That is:
\[
\frac{c - 2}{c - 3} = 2
\]
Now, solve for \( c \).

Step 1: Multiply both sides by \( c - 3 \) to eliminate the denominator:
\[
c - 2 = 2(c - 3)
\]

Step 2: Expand the right-hand side:
\[
c - 2 = 2c - 6
\]

Step 3: Rearrange the equation to isolate \( c \):
\[
c - 2c = -6 + 2
\]
\[
-c = -4
\]
\[
c = 4
\]

Thus, \( c = 4 \). 

Since 4 is within the interval \([-3, 4]\), this solution is valid.

### Problem 2: Function \( f(x) = \frac{1}{2}(x - 1)^2 - 2 \)
You are asked to find \( c \) such that \( f(c) = -2 \) in the interval \([-1, 4]\).

The function is given by:
\[
f(x) = \frac{1}{2}(x - 1)^2 - 2
\]
We need to find \( c \) such that:
\[
f(c) = -2
\]
That is:
\[
\frac{1}{2}(c - 1)^2 - 2 = -2
\]

Step 1: Add 2 to both sides:
\[
\frac{1}{2}(c - 1)^2 = 0
\]

Step 2: Multiply both sides by 2 to eliminate the fraction:
\[
(c - 1)^2 = 0
\]

Step 3: Take the square root of both sides:
\[
c - 1 = 0
\]

Step 4: Solve for \( c \):
\[
c = 1
\]

Thus, \( c = 1 \). 

Since 1 is within the interval \([-1, 4]\), this solution is valid.

---

### Summary:
- For \( f(x) = \frac{x - 2}{x - 3} \), \( c = 4 \).
- For \( f(x) = \frac{1}{2}(x - 1)^2 - 2 \), \( c = 1 \).

Let me know if you would like any additional details or have any questions!

Here are some related questions:
1. What happens if \( f(x) = 0 \) for these functions? 
2. How do these functions behave as \( x \to \infty \)?
3. Can you find the asymptotes of \( f(x) = \frac{x - 2}{x - 3} \)?
4. How do you determine the vertex of the quadratic function \( f(x) = \frac{1}{2}(x - 1)^2 - 2 \)?
5. What is the domain and range of each function?

**Tip:** Always check the domain of the function before solving for \( x \), especially when dealing with fractions or square roots.

Consider the functions and find c in the given intervals:
1. f(x) = (x - 2)/(x - 3), such that f(c) = 2 in [-3, 4]
2. f(x) = 1/2 * (x - 1)^2 - 2, such that f(c) = -2 in [-1, 4]

Learn how to solve rational and quadratic functions by finding the value of c in specific intervals. This detailed breakdown covers the steps to solve f(x) = (x - 2)/(x - 3) and f(x) = 1/2 * (x - 1)^2 - 2, ideal for high school students (Grades 9-12).

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