Section 6.7 – Inverse relations and Inverse Functions
Objectives:
- Determine the inverse of a function numerically, graphically and algebraically.
- Recognize and identify one-to-one functions and sketch the graph of an inverse.
- Find the domain of the inverse, stating any restrictions “inherited” from the original function and any
restrictions needed on the domain of the original function to guarantee that the inverse is a function.
- Use inverse notation, f 1 ( x)
- Verify that two functions are inverses of each other
Example 1:
1) Graph f ( x)  2 x  4
2) Reflect f(x) over the line y = x.
3) Complete a table of values for both graphs.
F(x)
X
Y
-2
0
2
reflection of F(x) over y = x
X
Y
-2
0
2
4) What is the equation of the reflection? __________________
5) What relationships do you notice between the functions?
4 properties of inverse relations
-
Inverse operations in reverse order
Reflection over y = x
x and y values switch
domain and range switch
Using the following graphs, sketch the inverse relation and determine if the inverse is a function.
Graph A
Graph B
Graph C
Write the equation for the inverse relationship and state the domain and range.
Graph A: __________________
Graph B: ____________________ Graph C: _________________
Without drawing the inverse relation, how could you tell if the inverse for the graphs below would be a
function?
Functions whose inverses are also functions are said to be one-to-one. One-to-one means that every x
corresponds to a unique y and every y corresponds to a unique x. Any function which is one-to-one will
have an inverse which is also a function.
Look again at the graphs above, which are one-to-one?
When the inverse of a function is also a function, we use the following notation to denote the inverse
relationship: f 1 ( x) Note that this notation refers to the inverse of the function f(x) and the -1 is simply
a notation and not an operation.
For the graph below:
a) Write an equation for the graph using f(x).
b) State the Domain and Range of f(x).
c) Sketch the inverse of f(x).
d) State the Domain and Range of the inverse.
e) Write an equation for the inverse relationship, use inverse function notation if appropriate.
a) __________________
a) __________________
a) __________________
b) __________________
b) __________________
b) __________________
c) SKETCH
c) SKETCH
c) SKETCH
d) __________________
d) __________________
d) __________________
e)
e) ___________________
e) __________________
a) __________________
a) __________________
a) __________________
b) __________________
b) __________________
b) __________________
c) SKETCH
c) SKETCH
c) SKETCH
d) __________________
d) __________________
d) __________________
e)
e) ___________________
e) __________________
__________________
__________________
One way to prove that a function is one-to-one or that two functions are inverses of each other is to use
the inverse composition rule.
Show algebraically that f ( x)  x 3  1 and
g ( x)  3 x  1 are inverse functions.
Show (graphically) that f ( x)  x  3 has an inverse function and find a rule for f 1 ( x) . State the domain
and range for both functions.