Section 12.1 Notes – Algebra 2
Inverse Functions and Relations
Name: _____________________________
Period: _________
Goal: Be able to write the equation of the inverse of a relation, determine whether the graph of a
relation is symmetric with respect to the line y  x , and find the equation for an inverse function.
Vocabulary: Interchanging (switching) the x and y variables in an equation produces an ___________.
1. f  x   2x 10
2. f  x   4x  3
3. T  x   1 x
4. f  x   x  2
2
5. Are y  x 2  4 and y   x  4  inverse functions?
Ordered Pairs: Interchange the first and second coordinates (all possible y-values become x-values).
 The range (output/y-values) of f is the domain (input/x-values) of f 1
 The domain (input/x-values) of f is the range (output/y-values) of f 1
6. Find the inverse of the relation
 5, 1 ,  7, 1 , 5, 2  , 7, 2  ,  5, 3 , 7, 3 .
7. Find the inverse of the relation
 2, 3 ,  4, 3 ,  2, 8  .


Let A  1, 0  ,  2, 4  , 1, 3 .
Is A a function? ___________________________________________________________
Is the inverse of A a function? ________________________________________________
o Relations that are function may have inverses that are not functions.
o Relations that are not function may have inverses that are functions.
Graphs of Inverses
8. f  x   x  3
x
y
x
y
9. f  x   x 2  2
x
y
x
y
10. f  x   x 2  4
x
y
x
y
Composition Functions and its Inverse








11. Find the composition functions of f g  x  and g f  x  when f  x   2x and g  x   x 2  1.
12. Find the composition functions of f g  x  and g f  x  when f  x   2x  3 and g  x   x  4 .




What happens if we find f f 1  x  and f 1 f  x  ?
11.
f  x   4x  5
RECAP!
Interchange the variables _________________ of a function f  x  to find the inverse function f 1  x  .
o Range of f is domain of f 1 .
o Domain of f is range of f 1 .
The inverse of a function is _______________ a function.
o If not, it is called a relation.
The graph of a function and its inverse are reflection across the line _____________.
o y  x is the axis of symmetry




Two functions, f  x  and g  x  , are _____________ if f g  x   x and g f  x   x .