We are given a piecewise function: $f(x) = \begin{cases} x-1, & x < n \\ -x+4, & x \ge n \end{cases}$ We need to find the range of $f$ when $n=5$, and also determine if changing the value of $n$ changes the range of $f$.

1. Problem Description

We are given a piecewise function:

$f(x) = \begin{cases} x-1, & x < n \\ -x+4, & x \ge n \end{cases}$

We need to find the range of $f$ when $n=5$, and also determine if changing the value of $n$ changes the range of $f$.

2. Solution Steps

a. If $n = 5$, the piecewise function becomes:

$f(x) = \begin{cases} x-1, & x < 5 \\ -x+4, & x \ge 5 \end{cases}$

For $x < 5$, $f(x) = x - 1$. As $x$ approaches 5 from the left, $f(x)$ approaches $5 - 1 = 4$. So, $f(x) < 4$ for $x < 5$.

For $x \ge 5$, $f(x) = -x + 4$. When $x = 5$, $f(5) = -5 + 4 = -1$. As $x$ increases, $f(x)$ decreases, and it tends towards $-\infty$. So, $f(x) \le -1$ for $x \ge 5$.

Combining these two intervals, we have $f(x) < 4$ and $f(x) \le -1$. This means the range of $f$ is $(-\infty, -1] \cup (-\infty, 4)$, which is $(-\infty, 4)$.

Hence, the range is $f(x) < 4$.

b. To determine if changing $n$ affects the range, we need to analyze the function.

For $x < n$, $f(x) = x-1$. The range is $f(x) < n-1$.

For $x \ge n$, $f(x) = -x+4$. When $x=n$, $f(n) = -n+4$.

As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$. So the range is $f(x) \le -n+4$.

The range of the entire function is $(-\infty, n-1) \cup (-\infty, -n+4]$, which equals to $(-\infty, max(n-1, -n+4)]$.

We need to solve $n-1 = -n+4$ to determine the value of $n$ for which the two ranges connect.

$2n = 5$, so $n = \frac{5}{2} = 2.5$.

If $n > 2.5$, then $n-1 > -n+4$. So, the range is $f(x) < n-1$.

If $n < 2.5$, then $n-1 < -n+4$. So, the range is $f(x) < -n+4$.

If $n=2.5$, then the range is $f(x) < 1.5$.

Regardless of the value of $n$, the range will be $(-\infty, max(n-1, -n+4)]$.

Since $n-1$ is increasing in $n$ and $-n+4$ is decreasing in $n$, the range is dependent on the value of $n$.

3. Final Answer

a. If n = 5, the range of f is $f(x) \le -1$ or $f(x) < 4$. Thus $f(x) < 4$.

b. Yes, changing the value of n changes the range. The upper bound of the range changes with n. If $n > 2.5$, the upper bound is $n-1$. If $n < 2.5$, the upper bound is $-n+4$.