Finally, to indicate that this is the inverse function of f(x) we will replace y with f-1(x).
f-1(x)=x4+1
b Let's begin with finding f(f-1(x)). To do so we will substitute x=f-1(x) into our function, f(x)=x−14. Let's remember that we found f-1(x)=x4+1 in Part A.
x=f-1(x)⇒x=x4+1
Therefore, we will substitute x=x4+1 into the function f(x).
Moreover, since the inputs and also the outputs are the same for f(f-1(x)) and f-1(f(x)), they are called identity functions.
c We will examine the domain and range of the functions f and f-1. Let's begin with f.
f(x)=x−14
Note that f(x) is a rational function which is only undefined where the denominator is zero.
x−1=0⇒x=1
Since its denominator is 0 at x=1, its domain is all real numbers except 1.
Now, let's recall that the range is the set of all outputs of a function. Notice that the division of a number by another number never gives 0. Therefore, the range of the function is the set of all real numbers except 0.
Since x4 is also a rational function, it is undefined where x=0. Therefore, its domain is all real numbers except 0. For the range, since x4 never equals 0 we cannot get 1 as a result. Therefore, the range of the inverse function is the set of real numbers except 1.
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