Question

Identify which of the twelve basic functions listed below fit the description given.

Two functions that are bounded above and below.
A) y = x, y = x²
B) y = eˣ, y = ln(x)
C) y = sin(x), y = cos(x)
D) y = 1 + e⁻ˣ

Answer 1

Option C, consisting of the functions y = sin(x) and y = cos(x), fits the description of functions bounded both above and below, with their values ranging between -1 and 1.

Step-by-step explanation:

The question asks to identify two functions that are bounded above and below out of the ones listed. A function is bounded above if there is some value that it never exceeds and bounded below if there is a value it never goes under. Reviewing the options, option C: y = sin(x), y = cos(x) fits this description. The sine and cosine functions have values that range from -1 to 1; they never go beyond these bounds. Hence, they are both bounded above by 1 and bounded below by -1.

The functions that are bounded above and below are:

y = x

y = x²

These functions have both a minimum and a maximum value. For example, the function y = x is bounded above by the line y = x² and below by the x-axis. Similarly, the function y = x² is bounded above by the line y = x and below by the x-axis.