Which functions are always increasing? (There's more than one)

Question

Cubic

Cube Root

Square Root

Absolute Value

Log

Exponential

Quadratic

Rational

Answer 1

Simply providing you with the answer would not help you understand how these functions operate. I suggest graphing each of these functions on a calculator or by hand as a functions of x and notice the pattern of behavior as x increases.

For example

Cubic function can be graphed as x³

Cube root function can be graphed as x^1/3 and so on.

You can manually draw these functions. To do so start off by plugging in small values for x and increasing the values to see the pattern of behavior.

For example

f(x) = x³

f(1) = 1³ = 1

f(10) = 10³ = 1,000

f(100) = 100³ = 1,000,000

Answer 2

Proving that a function is always increasing (or decreasing) is in general not an easy task. I imagine that this is a conceptual question. First, try some test points. If a function seems to be increasing for several test points, then think about its graph to determine if its always increasing.

Examples:

1. For something like y(x) = |x|, y(-1) = 1 > y(0) = 0, but y(1) = 1>y(0), so this function is obviously not increasing.
2. For y(x) = x^3, you should know what the graph looks like (it is always increasing).
3. A rational function might have asymptote(s). Try to think of such a function that changes sign across an asymptote (for example y(x) = 1/x). Clearly, this is not an increasing function.

Hope this helps!

Answer 3

This question was answered here yesterday if you want to find it! Just go through the past questions