SOLUTION: show that the given function is one-to-one and find its inverse. Check your answers algebraically and graphically. Verify that the range of f is the domain of f^−1 and vice-vers

Algebra ->  Finance -> SOLUTION: show that the given function is one-to-one and find its inverse. Check your answers algebraically and graphically. Verify that the range of f is the domain of f^−1 and vice-vers      Log On


   

Question 1135135: show that the given function is one-to-one and find its inverse. Check your
answers algebraically and graphically. Verify that the range of f is the domain of f^−1 and vice-versa.

f(x) = 3√x−1 − 4
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

show that the given function is one-to-one and find its inverse. Check your
answers algebraically and graphically. Verify that the range of f is the domain of f^−1 and vice-versa.
f%28x%29+=+3sqrt%28x-1%29+-+4+
It is a 1-1 function if it passes both the vertical+line test and the horizontal line test.
if f%28a%29+=f%28b%29+ implies that a=b, then f%28x%29 is 1-1
3sqrt%28a-1%29+-+4=+3sqrt%28b-1%29+-+4
3sqrt%28a-1%29+=+3sqrt%28b-1%29
sqrt%28a-1%29+=+sqrt%28b-1%29
a-1+=+b-1
a+=+b
so, your function is injective (one-to-one)
inverse:
f%28x%29+=+3sqrt%28x-1%29+-+4+......f%28x%29+=+y+
y=+3sqrt%28x-1%29+-+4+......swap x and y
x=+3sqrt%28y-1%29+-+4+.......solve for y
x%2B4=+3sqrt%28y-1%29++
%28x%2B4%29%2F3=+sqrt%28y-1%29++.........square both sides

%28%28x%2B4%29%2F3%29%5E2=+%28sqrt%28y-1%29%29%5E2++
%28x%2B4%29%5E2%2F9=+y-1++
y=x%5E2%2F9%2B8x%2F9%2B16%2F9%2B1++
y=x%5E2%2F9%2B8x%2F9%2B16%2F9%2B9%2F9++
y=x%5E2%2F9%2B8x%2F9%2B25%2F9++
y=%281%2F9%29%28x%5E2%2B8x%2B25%29++

f%5E-1%28x%29=%281%2F9%29%28x%5E2%2B8x%2B25%29+




Answer by greenestamps(13206) About Me  (Show Source):
You can put this solution on YOUR website!


The solution by tutor @MathLover1 appears to be complete and correct (I didn't look at all the details). But for many relatively simple functions, there is an easier way to find the inverse, based on the idea that an inverse function has to "get you back where you started". Finding the function that gets you back where you started means looking at the operations the function performs on the input and performing the opposite operations in the opposite order.

In this example, the operations performed on the input by the function are

(1) subtract 1
(2) take the square root
(3) multiply by 3
(4) subtract 4

To get you back where you started, the inverse function has to

(1) add 4
(2) divide by 3
(3) square
(4) add 1

So the inverse function is

f%5E%28-1%29%28x%29+=+%28%28x%2B4%29%2F3%29%5E2%2B1

Looks a lot different than the inverse function shown by the other tutor; but they are equivalent.