Question

Find Inverse for f(x)=7
f^-1 (x)

Answer 1

f^-1 (x) = 1/0

Answer 2

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:

Answer 3

now it will be f^-1 (x) = 1/0

Answer 4

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!"

Answer 5

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....