Inverse Functions

  • From: Kelly Biddington
  • Date: 22 April 1999
  • Subject: Inverse Functions

The function f is given by   f:x --> ln(4 - 2x),   x<2

  1. Find an expression for f^-1(x)
  2. State the range of f^-1
  3. Sketch the graph of y = f^-1(x)

When it comes to e and ln, I become all quivery at the knees!!!

Maths Help suggests:

Just as addition and subtraction are inverse processes (each 'undoes' the other),
and squaring and square rooting are inverse proceses,
ex   and   ln x   are inverse processes.

To solve your problem, let   y = f(x)   then make x the subject:

y = ln(4-2x)  ->  x = 2 - 0.5e^y

This function of y is the form of the inverse of the original function of x.

Write the inverse function as   f^-1:  x = 2 - 0.5e^y

The range of the inverse function f-1(x) is the domain of f(x), namely   x < 2.

To make a successful sketch of the inverse function, you should work out
the y-intercept and get the general shape right:

Sketch of inverse function

Convince yourself that this can be built up from the basic graph of y = ex as follows:

REMEMBER that a useful way to get the graph of an inverse function is to reflect the graph of the original function in the line y=x. We have not done that here for two reasons.
Firstly, the graph of f(x) is itself not so straightforward to sketch (it needs a series of transformations of the basic graph y = ln(x)). And secondly, in this particular question, the graph of f(x) and its inverse overlap quite considerably. We suggest you investigate them using a graphics calculator or computer graph plotter.

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