If f is an increasing function, what can be said about f(a) and f(b) when a < b?

If f is an increasing function, what can be said about f(a) and f(b) when a < b?

• A. f(a) > f(b)
• B. f(a) < f(b)
• C. f(a) = f(b)
• D. f(a) <= f(b)

Answer: (B)

The key idea is that increasing functions preserve the order of inputs in their outputs. When one input is smaller than another, the output cannot run past it. If the function is strictly increasing, each time you move to a larger input, the output must also move to a larger value. So with a < b, we have f(a) < f(b). In many contexts, “increasing” is understood to mean strictly increasing, which is why the statement uses a strict inequality. If the function were only non-decreasing, we could only say f(a) ≤ f(b).

The key idea is that increasing functions preserve the order of inputs in their outputs. When one input is smaller than another, the output cannot run past it. If the function is strictly increasing, each time you move to a larger input, the output must also move to a larger value. So with a < b, we have f(a) < f(b). In many contexts, “increasing” is understood to mean strictly increasing, which is why the statement uses a strict inequality. If the function were only non-decreasing, we could only say f(a) ≤ f(b).