Teaching Notes 8.10: Finding the Inverse of a Function
Finding the inverse of a function involves switching the values ofxandyin the equation and
solving fory. The idea of switching values often confuses students.- Ask your students to consider the functionf(x)= 2 x. Explain that the ordered pairs (−1,
−2), (0, 0), (1, 2), and (3, 6) are solutions toy= 2 x.Theinverseoff(x), denoted asf−^1 (x),
is the function whose graph contains the points (−2,−1), (0, 0), (2, 1), and (6, 3). Note that
the coordinates (x,y)off(x)arethe(y,x) coordinates off−^1 (x). - Review the information and examples on the worksheet with your students. Use the first
example to verify that (−2,−1), (0, 0), (2, 1), and (6, 3) are the solutions tof−^1 (x)=
x
2.
Note that the domains of the functions are restricted so that the functions are defined.EXTRA HELP:
The inverse you found is correct iff(f−^1 (x))=1.ANSWER KEY:
(1)f−^1 (x)=1
7
x (2)g−^1 (x)=−1
4
x (3)h−^1 (x)=^3√
x(4)I−^1 (x)=
x+ 10
3
(5)F−^1 (x)=^3√
x+ 9
2
(6)k−^1 (x)=x− 2
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(Challenge)Jeffrey is incorrect.f−^1 (x)=±√
xis not a function because 4, for instance, is
------------------------------------------------------------------------------------------paired with 2 and−2.292 THE ALGEBRA TEACHER’S GUIDE