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A student says that all linear functions are either increasing or decreasing. Do you agree? Explain.

Choose the correct answer below.

A. Yes. Linear functions always have [tex]\((-∞,∞)\)[/tex] as their domain and range, so they are always increasing or decreasing.
B. No. Linear functions can also be constant, which means neither increasing nor decreasing.
C. No. Linear functions can be both increasing and decreasing.
D. Yes. Linear functions always have an x- and y-intercept, so they are always increasing or decreasing.

Sagot :

GinnyAnsweridn
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.
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