4.1 Inverse Functions
I. Basic Concepts
A. Relation: a correspondence between two sets - a set of inputs called the domain & a set
of outputs called the range.
B. Function: a special type of relation where every input has one and only one output.
A function will pass a Vertical Line Test.
C. One-to-one Function: a special type of function where every input has one & only one
output and every output has one & only one input. 1-1 functions pass a Vertical Line Test
and a Horizontal Line Test.
D. The Inverse of a Function
Informal Definition: The inverse of a function undoes what the original function did.
Formal Definition: Functions f and g are inverse functions if and only if
(f o g)( x) = f (g (x)) = x and (g o f) (x) = g (f (x)) = x.
Caution: f inverse does not equal 1 over f
II. Finding the Formula for the Inverse of a Function
Before you do anything else, determine if the given function is 1-1 by making an accurate “quick
sketch” and then doing a VLT and a HLT. If the function is not 1-1, it does not have an inverse.
If it is 1-1, find its inverse.
A. Finding an Inverse Algebraically
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve for y.
4. Replace y with f -1(x).
a. Sketch the graph of and use the graph to determine whether the function is 1-to-1.
Since the graph of f (x) passes both a vertical line test
and a horizontal line test, it is a 1-to-1 function and
it does have an inverse.
b. If the function is 1-to-1, algebraically find a formula for the inverse.
B. Finding an Inverse Informally
To undo what the original function did, the inverse function must do the opposite
operations in the opposite order.
Find the inverse of g (x) = 7x + 2 by thinking about the operations of the function and then undoing them.
Check your work algebraically.
Since the graph of g (x) is an oblique line, it passes both a vertical and a horizontal line test. Therefore g
does have an inverse.
Function g first multiplies x by 7 and second adds 2. To undo this, the inverse function must
first subtract 2 and second divide by 7. Therefore,
III. Characteristics of Inverse Functions
A. Only one-to-one functions have inverses.
B. The domain of f is the range of f -1 and the range of f is the domain of f -1.
C. If you make a t-chart for f and then use those y values as the x values to make a t-chart
for f -1, the two t-charts will be mirror images of one another, i.e., their x values and y
values will be reversed.
D. f and f -1 are symmetric around the y = x line.
E. If f and g are inverse functions, (f o g) (x) = f (g (x)) = x and (g o f) (x) = g (f( x)) = x.
Algebraically find the inverse of the given one-to-one function, f (x) = 3x2 – 5, x ≥ 0.
Give the domain and range of f and f -1. Then graph f and f -1 on the same set of axes.
Note: Without the restriction on the domain, f would not be a 1-to-1 function.
Note: Because of the restriction on the domain of f, x ≥ 0, f -1 has a restriction
on its range, y ≥ 0, that is how we know the square root is positive and
Domain of f = Range of f -1 = [0, ∞ ) and Range of f = Domain of f -1 = [–5, ∞ )
IV. Verifying Algebraically that Two Functions are
If (f o f -1 )(x) = f (f -1(x)) = x and (f -1 o f )(x)= f -1(f(x)) = x, then f and f -1 are inverse functions
For the function f, use composition of functions to show that f -1 is as given.
Since f (f -1(x)) = x and f -1(f(x)) = x, f and f -1 are inverse functions.