Prove that f is invertible and find (f^-1)'((1)/(2) ?

If f(x) = cos x for all x in (0, (pi)/(2)), prove that f is invertible and find (f^-1)'((1)/(2)).

1 Answer
Mar 31, 2018

arccos(1/2)= pi/3

Explanation:

To know if a relation is invertible we must know the criteria for being invertible. The criteria are as follows:

❈ It must be a function, meaning that each value of x maps to only 1 y value.

❈ The inverse function must also be a function, meaning that each value of x maps to only 1 y value.

❈ The inverse function must be a reflection of y=f(x) in the line y=x. (you can test this using http://desmos.com/calculator)

❈ Must satisfy f(f^{-1}(x)) = x and f^{-1}(f(x)) = x

First of all, as f(x) = cosx is a function that maps to only 1 value for each x value, it is invertible for at least a certain range.

The inverse function of f(x)=cosx is f^{-1}(x)=arccos(x), by definition.

However since cosx is a repeating function it is a many-to-one function, meaning that several x values will give the same y value. This means that the inverse function will be a one-to-many relation (since y and x switch places), and therefore not a function. This problem can be overcome by restricting the domain, which has already been done: x in (0, pi/2). The function is defined for this domain.

arccos(1/2)= pi/3

Now to prove that it is invertible we want to test it using the following equations:

f(f^{-1}(x)) = x and f^{-1}(f(x)) = x

cos(arccos(1/2)) =1/2
arccos(cos(pi/3)) = pi/3