Question

f(x) = cos(x) defined on f : [a,b] $\to$ [-1,1] has its inverse then which of the following is correct?

• A. $a=-\pi ;b=\pi$
• B. $a=0;b=\pi$
• C. $a=0;b=2\pi$
• D. None of these

Answer 1

The correct option is B $a=0;b=\pi$

Well, we know that for a function to have an inverse, the function should be one- one onto. We can see the co- domain given in the question is same as the range of cos(x).
So now we only have to see that interval in which cos(x) is one - one. It is very simple, just draw horizontal lines, if any line cuts the curve at more than one point then it’s a many one function whereas if no line cuts more than once then it’s a one - one function.
Let’s check the options given one by one.
$a=-\pi ;b=\pi$

We can see that a horizontal line shown is cutting the curve at two points. So this options can’t be correct.

$b.a=0;b=\pi$

We can see that we cannot find any line which would cut the graph more than once in the interval given. Actually the function is a strictly decreasing function in the interval. Thus we can define the inverse for the function in this interval. And thus it is the answer.
$C.a=0;b=2\pi$

In this interval also we can see that there are many lines we can get which will cut the graph at more than one points.