]> Exercise 1.8

## Exercise 1.8

For what values can you define the inverse of the function $cos ⁡ ( sin ⁡ x )$ . (Hint: set $f = cos ⁡ ( sin ⁡ x )$ look at its inverse and figure out the answer.)

Solution:

The inverse function $f$ , with $f = cos ⁡ ( sin ⁡ x )$ can be defined at any argument that is a value for $cos ⁡ ( sin ⁡ x )$ . Since $sin ⁡ x$ takes values between -1 and 1, and cosine is an even function, its values lie between $cos ⁡ 0$ (which is 1) and $cos ⁡ 1$ , which is .540302306, more or less. Therefore this inverse function can be defined between this value and 1. In that domain its value must be chosen to be one from among the various arguments for $cos ⁡ ( sin ⁡ x )$ which have that value. (for example, you could choose a value between 0 and $π 2$ .)