Module 2 
2.4
Study Notes (for Text section 4.1)
In order for a function’s inverse relation to also be a function, it must be possible to determine the argument completely from the result, so there must be only one input giving each possible output. Functions with this property are said to be one-to-one
Q1
You should be able to test whether a function
is one-to-one either algebraically as in the text’s Example 1, or graphically
by the “Horizontal Line Test” (HLT) as in the text’s Example 2. If a function
is always increasing, then as we read left to right its graph is always rising
so it never hits the same value twice and so is one-to-one. The same applies if
the function is always decreasing. Can you draw a function graph that is not
either always increasing or always decreasing but is still one-to-one? (see answer #1)
Graphs of Inverse Functions
Q2
Since the equation
just means that
,
the graph of
comes from
just by interchanging x and y , or in other
words by reflecting across the diagonal
.
We don’t even need a formula in order to do this. For example, the function
graph below passes the HLT so the function f
has an inverse function which you should be able to graph without doing any
algebra. (see answer #2)
Finding Inverse Functions
If and f
is one-to-one, then
.
So if we can solve for x in the equation ,
then the result must be a formula for ,
and to get
we just have to replace y by x in the resulting
formula.
It also works to go the other way, since if
then
,
so solving for y in terms of x after switching should lead to the same result.
Try this second approach with the text’s Example#3 on page 284. (see answer#3)
Further Practice
Check your understanding, and practice for speed, by working through some of the Exercises on pages 288-290 of the text (at the end of Section 4.1).
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort)
As a minimum, you should try ##1, 7, 11, 17, 21, 27, 31, 37, 41, and 53.