Inverse Functions
Remember that inverse operations undo each other, like
addition/subtraction, multiplication/division, or exponents/roots. So, an example
x
of a pair of inverse functions would be f(x) = 2x and g(x) = . In f, we’re
2
multiplying by 2 and in g, we’re dividing by 2 – inverse operations! Let’s plot
points for these two lines and examine the results. For f(x), I’m going to pick the
basic -2, -1, 0, 1, 2.
x
f(x) = 2x
g(x) =
What do you notice about
2
the “x” and “y” values in
x y plug this in here
x y
f(x) compared to the “x”
-2
and “y” values in g(x)?
-1
____________________
0
____________________
1
2
Another neat thing to know about inverses is that if you graph a pair of
inverse functions, you get a mirror image about the line y = x. Also, if the inverse
functions cross each other, it’s going to be on the line y = x.
y=x
If we take a function f(x) in general, the inverse of it (called “f -1(x)”) will
have all of the inverse operations of the original function. So, if f(x) involves
multiplying by 3 and adding 5, the inverse f -1(x) will involve dividing by 3 and
subtracting 5. The process of finding an inverse function for a given f(x) involves
the idea we noticed with the points above: the “x” and “y” values switch places.
So, using this idea, let’s find the inverse of f(x) = 3x – 5.
Rewrite f(x) = 3x – 5 as y = 3x – 5.
My new equation is x = 3y – 5 Å we just switched the “x” and “y”. Now, solve
back for “y” by adding 5 and dividing by 3.
So, f -1(x) = ________________.
Simply put, the process is
a) Switch “x” and “y”
and
b) Solve back for the new “y”.
Here are a few more examples. Given f(x), find f -1(x).
3x − 6
7x
1) f(x) =
2) f(x) =
+5
4
9