Question

a piecewise - defined function is shown below.
$f(x)=\begin{cases}x - 1, & xlt n\\-x + 4, & xgeq nend{cases}$
a. if $n = 5$, what is the range of $f$?
b. does changing the value of $n$ change the range? explain.
a. if $n = 5$, the range of $f$ is $square$.
(type an inequality or a compound inequality )

Explanation:

Step1: Substitute n=5 into f(x)

$f(x)=

$$\begin{cases} x-1, & x<5 \\ -x+4, & x\geq5 \end{cases}$$

$

Step2: Analyze first piece's range

For $x<5$, $x-1 < 5-1=4$, so $f(x)<4$

Step3: Analyze second piece's range

For $x\geq5$, $-x+4 \leq -5+4=-1$, so $f(x)\leq-1$

Step4: Combine the two ranges

The overall range is all real numbers less than 4, since the second piece's values are a subset of $f(x)<4$.

Step5: Address part b logic

The first piece $x-1$ can take all values less than $n-1$, the second piece $-x+4$ can take all values less than or equal to $-n+4$. The maximum value of the entire function is the larger of $n-1$ and $-n+4$. Solve $n-1=-n+4$: $2n=5$, $n=\frac{5}{2}$. For $n<\frac{5}{2}$, $n-1<-n+4$, so the range is $f(x)<-n+4$; for $n>\frac{5}{2}$, $-n+4

Answer:

a. $f(x) < 4$
b. Yes, changing the value of $n$ changes the range. When $n$ changes, the upper bound of the function's output changes: for $n<\frac{5}{2}$, the range is $f(x)<-n+4$; for $n>\frac{5}{2}$, the range is $f(x)