Question

Determine whether the function is one-toone on its entire domain and therefore has an inverse function. $$ f(x)=\sin \frac{3 x}{2} $$

Short answer

Yes, the function \(f(x) = \sin \frac{3x}{2}\) is one-to-one on the specific domain [0, \(\frac{4\pi}{3}\)]. Hence, it has an inverse on this domain.

Step-by-step solution

Determine the period of the function

Usually a sine function has a period of \(2\pi\). The period is changed according to multiplying the variable inside the sine function. Hence, the period of the given function \(f(x) = \sin \frac{3x}{2}\) would be \(\frac{2\pi}{3/2} = \frac{4\pi}{3}\). Thus, if we take the domain to be just one period, such as [0, \(\frac{4\pi}{3}\)], the function would be one-to-one in this domain.

Specify the conditions for one-to-one

If the function repeats its values within the period, then it's not one-to-one, otherwise it is. A periodic function like sine repeats its values after every period. However, within one period (from 0 to the end of one period) the sine function does not repeat any of its values.

Check the condition

As we can see that within one period [0, \(\frac{4\pi}{3}\)], our function \(f(x) = \sin \frac{3x}{2}\) does not repeat any of its values. Therefore, this function is one-to-one within the specified domain.