SOLUTION: please explain: Determine whether f(x)=6/(x+5) is one-to-one. If it is one-to-one, find a formula for the inverse.

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Question 1140384: please explain:
Determine whether f(x)=6/(x+5) is one-to-one. If it is one-to-one, find a formula for the inverse.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

In order to show that f%28x%29 is one-to-one, we need to show that if f%28x%5B1%5D%29=f%28x%5B2%5D%29, then x%5B1%5D=x%5B2%5D.

In other words, we need to show that each y value ONLY has ONE unique x value mapping to it.
In this case, we're assuming that x%5B1%5D and x%5B2%5D are two different values that map to the same y+}value (but as you'll see, they are in fact the same value)

f%28x%5B1%5D%29=f%28x%5B2%5D%29 Start with the given equation
6%2F%28x%5B1%5D%2B5%29=6%2F%28x%5B2%5D%2B5%29 ...divide by 6
1%2F%28x%5B1%5D%2B5%29=1%2F%28x%5B2%5D%2B5%29 .......cross multiply
%28x%5B2%5D%2B5%29=%28x%5B1%5D%2B5%29 .......cancel 5
x%5B2%5D=x%5B1%5D
So we've shown that if f%28x%5B1%5D%29=f%28x%5B2%5D%29, then x%5B1%5D=x%5B2%5D. This means that f%28x%29 is one-to-one.
now, find a formula for the inverse:
f%28x%29+=+6%2F%28x+%2B+5%29+........recall that f%28x%29=y
y=+6%2F%28x+%2B+5%29+.......swap x and y
x=+6%2F%28y+%2B+5%29+.........solve for y
x%28y+%2B+5%29+=+6
y+%2B+5+=+6%2Fx
y++=+6%2Fx-5
f%5E-1%28x%29+=++6%2Fx-5


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+6%2F%28x%2B5%29%2C+6%2Fx-5%29+