## INCREASING AND DECREASING FUNCTIONS

Increasing and Decreasing Functions :

Here we are going to see, how to check if the function is increasing or decreasing from the graph.

Let us consider the following graph The domain of the function shown here is the interval [1, 6].

On the interval [1, 3], the graph of this function gets higher from left to right; thus we say that this function is increasing on the interval [1, 3].

On the interval [3, 6], the graph of this function gets lower from left to right; thus we say that this function is decreasing on the interval [3, 6].

Increasing on an interval :

A function f is called increasing on an interval if f (a) < f (b) whenever a < b and a, b are in the interval.

Decreasing on an interval :

A function f is called decreasing on an interval if f (a) > f (b) whenever a < b and a, b are in the interval.

Note :

We have to see the graph from left to right horizontally and vertically we have to see the graph from down to up.

## Increasing and Decreasing Functions - Examples

Question 1 :

The function f whose graph is shown here has domain [−1, 6]. (a) Find the largest interval on which f is increasing.

(b) Find the largest interval on which f is decreasing.

(c) Find the largest interval containing 6 on which f is decreasing.

Solution :

(a) As can be seen from the graph above, [1, 5] is the largest interval on which f is increasing.

(b) As can be seen from the graph above, [−1, 1] is the largest interval on which f is decreasing.

(c) As can be seen from the graph above, [5, 6] is the largest interval containing 6 on which f is decreasing.

Question 2 :

Shown below are the graphs of three functions; each function is graphed on its entire domain. (a) Is f increasing, decreasing, or neither?

(b) Is g increasing, decreasing, or neither?

(c) Is h increasing, decreasing, or neither?

Solution :

(a) The graph of f gets lower from left to right on its entire domain. Thus f is decreasing.

(b) The graph of g gets higher from left to right on its entire domain. Thus g is increasing.

(c) The graph of h gets lower from left to right on part of its domain and gets higher from left to right on another part of its domain. Thus h is neither increasing nor decreasing.

Question 3 :

Here f has domain [0, 4] and g has domain [−1, 5].

(i)  What is the largest interval contained in the domain of f on which f is increasing?

(ii)  What is the largest interval contained in the domain of g on which g is increasing? Solution :

(i)  The largest interval on which the function f increasing is [3, 4].

(i)  The largest interval on which the function g decreasing is [0, 3]. After having gone through the stuff given above, we hope that the students would have understood "Increasing and Decreasing Functions".

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