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Graphing Using Inverses
Pre-Calc. for AP Prep.
Date: ________
Goal:
Recall:
 For the inverse of a function to be a function, _______________________________
Ex: Determine if the inverse of each is a function.
b)
c) y  sin x
a)

If the function’s inverse is not a function, we can ____________________________
___________________________________________________________________
Ex: Graph f ( x)  x  3  1 and its inverse on the same axis. Then determine a
2
domain restriction that could be placed on f(x) so that its inverse is a function.
II. Trigonometric Functions and Their Inverses
A. Sketch a graph of the function f(x) = sin x. Determine if it has an inverse that is a
function.
B. On our calculators, there are inverse trigonometric function buttons, but the calculator
uses a restricted domain for sin x (this function is sometimes written Sin(x).
1. Use your calculator to graph f(x) = Sin 1 x . Give the domain and range of the graph.
How does this compare to the domain and range of y = Sin(x)?
2. Sketch a graph of y = cos x and y = tan x. Determine a domain restriction that
allows their inverses to be functions. Then graph the inverse on the same coordinate
plane as the original.
f ( x)  Cos x and
f 1 ( x)  Cos -1 x
g ( x)  Tan x and g 1 x   Tan -1 x 
III. Graphs of Logarithmic Functions
 The inverse of an exponential function is a _____________________________
o Do we need to limit the domain of the exponential function? _________
Ex: Graph f ( x)  e x and f 1 ( x)  ln x on the same coordinate plane. Then
give the domain and range for each.
2. Graph f ( x)  1  2 x3 , then graph its inverse on the same plane. What is the
equation of the inverse function?
IV. Using inverses of major families of graphs to graph relations.
 Sometimes relations (not functions) cannot expressed in such a way where the output
is written in terms of the input (i.e. ______________________________).
Ex: Graph each of the following.
x   y  2  3
2
x  2 y  1
x  2  3 y
HW: Handout.
x  1  23 y  3