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How do you find the inverse of # f(x) = (2x-1)/(x-1)# and is it a function?

1 Answer
May 14, 2016

Answer:

  • #f^-1(x)=(-x+1)/(2-x)#
  • yes, it is a function

Explanation:

Determining the Inverse Function
Given,

#f(x)=(2x-1)/(x-1)#

Substitute #y# for #f(x)#.

#y=(2x-1)/(x-1)#

Swap the #x# and #y#.

#x=(2y-1)/(y-1)#

Solve for #y#.

#x(y-1)=2y-1#

#xy-x=2y-1#

#2y-xy=-x+1#

Factor out #y# from the left side.

#y(2-x)=-x+1#

#y=(-x+1)/(2-x)#

Rewrite #y# as #f^-1(x)#.

#color(green)(|bar(ul(color(white)(a/a)color(black)(f^-1(x)=(-x+1)/(2-x))color(white)(a/a)|)))#

Determining Whether the Inverse Function Is a Function
Graphically, #f^-1(x)=(-x+1)/(2-x)# would look like:

graph{(-x+1)/(2-x) [-10, 10, -5, 5]}

In the graph above, you can see that the #x# and #y# values approach the vertical and horizontal asymptotes. Since it resembles that of an exponential graph, there is only one #y# value for an #x# value.

#:.#, the inverse function is a function.