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Let f : [0, ∞) → [0, ∞) and g : [0, ∞) → [0, ∞) be non-increasing and non-decreasing functions and h(x) = g[f(x)]. If f and g are differentiable for all points in their respective domains and h(0) = 0, then the value of h(3) is……

Let f : [0, ∞) → [0, ∞) and g : [0, ∞) → [0, ∞) be
non-increasing and non-decreasing functions and
h(x) = g[f(x)]. If f and g are differentiable for all
points in their respective domains and h(0) = 0,
then the value of h(3) is……

Grade:12

1 Answers

SAGAR SINGH - IIT DELHI
879 Points
11 years ago

Dear student,

Increasing Functions

A function is "increasing" if the y-value increases as the x-value increases, like this:

Increasing Function

It is easy to see that y=f(x) tends to go up as it goes along.

Flat ?

What about that flat bit near the start? Is that OK?

  • Yes, it is OK if you say the function is Increasing
  • But it is not OK if you say the function is Strictly Increasing (no flatness allowed)

Using Algebra

What if you can't plot the graph to see if it is increasing? In that case is is good to have a definition using algebra.

For a function y=f(x):

when x1 < x2 then f(x1) ≤ f(x2) Increasing
when x1 < x2 then f(x1) < f(x2) Strictly Increasing

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