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f^-1 (x) = 1/0
First, I recognize that f(x) is a rational function. Here's its graph: The restriction on the domain comes from the fact that I can't divide by zero, so x can't be equal to –2. I usually wouldn't bother writing down the restriction, but it's helpful here because I need to know the domain and range of the inverse. Note from the picture (and recalling the concept of horizontal asymptotes) that y will never equal 1. Then the domain is "x is not equal to –2" and the range is " y is not equal to 1". For the inverse, they'll be swapped: the domain will be "x is not equal to 1" and the range will be "y is not equal to –2". Here's the algebra: The original function: I rename "f(x)" as "y": Then I solve for "x =": I get the x-stuff on one side: Here's the trick: I factor out the x! Then I switch x and y: And rename "y" as "f-inverse"; the domain restriction comes from the fact that this is a rational function. Since the inverse is just a rational function, then the inverse is indeed a function. Here's the graph: Then the inverse is y = (–2x – 2) / (x – 1), and the inverse is also a function, with domain of all x not equal to 1 and range of all y not equal to –2. Find the inverse of f(x) = x2 – 3x + 2, x < 1.5 With the domain restriction, the graph looks like this: From what I know about graphing quadratics, the vertex is at (x, y) = (1.5, –0.25), so this graph is the left-hand "half" of the parabola. This half of the parabola passes the Horizontal Line Test, so the (restricted) function is invertible. But how to solve for the inverse? Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved The original function: f(x) = x2 – 3x + 2 I rename "f(x)" as "y": y = x2 – 3x + 2 Now I solve for "x =" by using the Quadratic Formula: 0 = x2 – 3x + 2 – y 0 = x2 – 3x + (2 – y) Since x < 1.5, then I want the negative square root: Now I switch x and y: And rename "y" as "f-inverse"; the domain restriction comes from the fact that this is a rational function. Then the inverse is given by:
now it will be f^-1 (x) = 1/0
When you get an answer of somthing/0, that's the universe's way of telling you "hey, there is no answer, this is a trick question!"
another way to look at this is that since f(x)=7 is a horizontal line, its inverse (if it has one) is a reflection of y=7 over the y=x line... but that will give you a vertical line... which is not a function....