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We are given a piecewise function \( g(x) \), and we need to evaluate it at three different points: \( g(-1) \), \( g(1) \), and \( g(5) \). The function is defined as:

\[
g(x) = \begin{cases} 
\frac{1}{4}x + 2 & \text{if } x \leq -1 \\
(x - 1)^2 - 1 & \text{if } -1 < x < 2 \\
-\frac{1}{4}x + 1 & \text{if } x \geq 2 
\end{cases}
\]

Let's go step by step.

### 1. Evaluate \( g(-1) \):

Since \( -1 \leq -1 \), we use the first case:
\[
g(-1) = \frac{1}{4}(-1) + 2 = -\frac{1}{4} + 2 = \frac{7}{4}
\]

### 2. Evaluate \( g(1) \):

Since \( -1 < 1 < 2 \), we use the second case:
\[
g(1) = (1 - 1)^2 - 1 = 0^2 - 1 = -1
\]

### 3. Evaluate \( g(5) \):

Since \( 5 \geq 2 \), we use the third case:
\[
g(5) = -\frac{1}{4}(5) + 1 = -\frac{5}{4} + 1 = \frac{-5}{4} + \frac{4}{4} = \frac{-1}{4}
\]

### Final Answers:
\[
g(-1) = \frac{7}{4}, \quad g(1) = -1, \quad g(5) = \frac{-1}{4}
\]

Would you like more details or have any other questions?

Here are 5 related questions to expand on the concept:

1. How would you describe the behavior of the function \( g(x) \) at \( x = 2 \)?
2. What is the domain and range of the function \( g(x) \)?
3. Can you graph the piecewise function and interpret the graph?
4. What happens to the values of \( g(x) \) as \( x \to -\infty \) and \( x \to \infty \)?
5. How would the function \( g(x) \) change if the second interval were \( -1 \leq x < 2 \)?

**Tip**: For piecewise functions, always carefully check which interval your input falls into before evaluating.

Suppose that the function g is defined, for all real numbers, as follows: g(x) = { 1/4x + 2 if x ≤ -1 (x-1)^2 - 1 if -1 < x < 2 -1/4x + 1 if x ≥ 2 Find g(-1), g(1), and g(5).

Learn how to evaluate a piecewise function g(x) at different points such as g(-1), g(1), and g(5). The function consists of three distinct expressions depending on the value of x. This problem covers key algebraic concepts related to piecewise functions and is suitable for high school students.

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