The student's statement is, "All linear functions are either increasing or decreasing." Let's examine this in detail.
Linear functions have the general form:
[tex]\[ f(x) = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. Increasing and Decreasing Functions:
- A linear function is increasing if its slope [tex]\( m \)[/tex] is positive ([tex]\( m > 0 \)[/tex]). As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] also increases.
- A linear function is decreasing if its slope [tex]\( m \)[/tex] is negative ([tex]\( m < 0 \)[/tex]). As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases.
2. Constant Functions:
- A linear function can also be constant if its slope [tex]\( m \)[/tex] is zero ([tex]\( m = 0 \)[/tex]). In this case, [tex]\( f(x) = b \)[/tex], a constant value. Here, the function is neither increasing nor decreasing, as it remains the same for all values of [tex]\( x \)[/tex].
Given this analysis, the assertion that "all linear functions are either increasing or decreasing" is incorrect because it overlooks the possibility of a constant linear function.
Therefore, the correct explanation is:
"B. No. Linear functions can also be constant, which means neither increasing nor decreasing."
This choice accurately addresses the fact that linear functions with a slope of zero are constant and thus do not fit into the categories of increasing or decreasing functions.
