SOLUTION: Can someone answer this for me? a) Show that the function f(x) = x/x-4 is a ont-to-one function? b) find the inverse of the function. Thank you.

Algebra ->  Functions -> SOLUTION: Can someone answer this for me? a) Show that the function f(x) = x/x-4 is a ont-to-one function? b) find the inverse of the function. Thank you.       Log On


   

Question 169138: Can someone answer this for me?
a) Show that the function f(x) = x/x-4 is a ont-to-one function?
b) find the inverse of the function.
Thank you.
Answer by jim_thompson5910(35256) 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


%28x%5B1%5D%29%2F%28x%5B1%5D-4%29=%28x%5B2%5D%29%2F%28x%5B2%5D-4%29 Plug in the function


x%5B1%5D%28x%5B2%5D-4%29=x%5B2%5D%28x%5B1%5D-4%29 Cross multiply


x%5B1%5D%2Ax%5B2%5D-4x%5B1%5D=x%5B2%5D%2Ax%5B1%5D-4x%5B2%5D Distribute


x%5B1%5D%2Ax%5B2%5D-4x%5B1%5D=x%5B1%5D%2Ax%5B2%5D-4x%5B2%5D Rearrange the terms.


Subtract x%5B1%5D%2Ax%5B2%5D from both sides. Notice how they cancel out on both sides.


-4x%5B1%5D=-4x%5B2%5D Simplify


cross%28%28-4%29%2F%28-4%29%29x%5B1%5D=cross%28%28-4%29%2F%28-4%29%29x%5B2%5D Divide both sides by -4. Once again, they cancel out on both sides.


x%5B1%5D=x%5B2%5D Simplify


So we've just 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.


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f%28x%29+=+x%2F%28x-4%29 Start with the given function


y+=+x%2F%28x-4%29 Replace f(x) with y


x+=+y%2F%28y-4%29 Switch x and y. The goal is to solve for y


x%28y-4%29+=+y Multiply both sides by y-4


xy-4x+=+y Distribute



-4x+=+y-xy Subtract xy from both sides.


-4x+=+y%281-x%29 Factor out the GCF y


-%284x%29%2F%281-x%29+=+y Divide both sides by 1-x to isolate y


So the answer is y=-%284x%29%2F%281-x%29+ which means that the inverse function is