SOLUTION: for the one-to-one function f(x)= 2/x-4 Find the inverse f^-1 Find the domain and range of f Find the domain and range of f^-1

Algebra ->  Functions -> SOLUTION: for the one-to-one function f(x)= 2/x-4 Find the inverse f^-1 Find the domain and range of f Find the domain and range of f^-1      Log On


   

Question 206027: for the one-to-one function f(x)= 2/x-4
Find the inverse f^-1
Find the domain and range of f
Find the domain and range of f^-1
Answer by mickclns(59) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+2%2Fx-4
To find the inverse function of a function, replace x by y and y (or f(x) ) by x and solve for y
 
x+=+2%2Fy-4      add 4 to each side
x%2B4+=+2%2Fy     change the left side to a fraction and flip both sides
1%2F%28x%2B4%29+=+y%2F2     multiply both sides by 2, then switch sides so y (the inverse of f(x) ) is on the left
f%5E-1%28x%29+=+y+=+2+%2F+%28x%2B4%29
 
Here is a graph of f (red) and its inverse (green, the vertical line at -4 should be ignored, it is not part of either function). Also, graphed is the line y=x ... a function and its inverse are always mirror images of one another across this line, which you can see for the green and the red here.
graph%28300%2C300%2C-10%2C4%2C-10%2C4%2C%282%2Fx%29+-+4%2C+2%2F%28x%2B4%29%2C+x+%29+
 
 
The domain of f is all real numbers except 0 (because x = x-0 is in the denominator makes f(x) undefined).
The range of f is all real numbers except -4 ... for -4 to be the value of y, 2%2Fx would have to be 0 which means x would have to be infinitely large, but no real number is infinitely large.
The domain of f%5E-1 is all real numbers except -4 (because x+4 = -4+4 is in the denomiator and makes f%5E-1 undefined).
The range of f%5E-1 is all real numbers except 0 ... for 0 to be the value of y, 2%2F%28x-4%29 would have to be 0 which means x-4 (and therefore x) would have to be infinitely large, but no real number is infinitely large.