Question

A graph of a function f is given. For what interval is f increasing?

A. (-1,3]
B. (-3,1)
C. (-3,1]
D. none of these

Answer 1

Hint: The function f is said to be increasing for a certain interval of x, if the value of y always increases with increase in the value of x belonging to that interval. Also, the slope of tangent to the curve at every point in these intervals is positive.

Complete step-by-step answer:
Let us first understand what is meant by increasing function.
A function is said to be increasing when the value of the function (dependent variable) increases when the value of the dependent variable increases.
Suppose we define a function $ y=f(x) $ . Then the function f is said to be increasing for a certain interval of x, if the value of y always increases with increase in the value of x belonging to that interval.
Also, when the function f is strictly increasing, its derivative with respect to x in that interval is always positive.
i.e. $ y'>0 $ .
Or we can also say that the slope of tangent to the curve at every point in these intervals is positive.
In the given graph, we can see that the function f is increasing for some interval of x and it's also decreasing for some interval of x.
With the above discussed points, we get that the function f is strictly increasing between $ x=-1 $ and $ x=3 $ .
Therefore, the function f is increasing in the interval (-1,3].
So, the correct answer is “Option A”.

Note: Sometimes, students may make mistakes in using the correct brackets.
When we write the interval as (-1,3], it means that the value $ x=-1 $ is excluded from the interval. This is indicated by the open bracket ‘(’ or ‘)’.
Whereas the value $ x=3 $ in included in the interval. This is indicated by a closed bracket ‘[’ or ‘]’ .
The tangent or derivative of the function at $ x=-1 $ is zero. Therefore, we cannot include it in the interval.