Question

Which of the following functions are Monotonically increasing functions throughout the domain?

• A. y = e^x
• B. y = x²
• C. y = [x] where, [] is greatest integer function.
• D. None of these

Answer 1

Answer: (A), (C)

The correct options are
A

y = e^x

C

y = [x] where, [] is greatest integer function.

We will use the graph of the functions to solve this.

y = e${}^x$

As we can see from the graph, the function increases its value as we increase x. The curve is always going up, never coming down. So we can say that this is a monotonically increasing function

(b.) y = x${}^2$

As we can see, the value of the function comes down to zero for negative values of x and increases from zero for positive values of x. Since the function is decreasing its value when we increase x, in the interval $(-\infty ,0)$, we will say that this function is not monotonically increasing.

(c.) y = [x] where, [] is greatest integer function.

As we can see, the function remains constant between two consecutive integers and increases its value in the other intervals. Since an increasing function can be constant, this is an example of monotonically increasing function. Monotonically increasing means never decreasing. Y = [x] is never decreasing

So the answers are A and C