Answers Inverse Functions Handout
1. Inverse Functions
You already know that exponential functions and logarithmic functions are inverses of
one another. Now, let’s see what is implied by the term “inverse”.
Given 𝒇 𝒙 = 𝟐𝒆𝟑𝒙 + 𝟏
(a) Find the inverse, 𝑓 !! 𝑥 . Show your work.
𝒙−𝟏
𝒍𝒏
𝒙
−
𝟏
𝟐
𝒙 = 𝟐𝒆𝟑𝒚 + 𝟏 → 𝒙 − 𝟏 = 𝟐𝒆𝟑𝒚 → 𝒍𝒏
= 𝟑𝒚 → 𝒔𝒐 𝒇!𝟏 𝒙 =
𝟐
𝟑
(b) Now fill in the table below for 𝑓 𝑥 and 𝑓 !! (𝑥).
𝒙
𝒇(𝒙)
𝒇!𝟏 (𝒙)
0
3
Undef.
1
41.171
Undef.
2
807.86
-0.231
3
16207
0
4
325511
0.13516
Is there a specific pair of points that stand out to you?
𝒇 𝒙 contains the point (0, 3) and 𝒇!𝟏 (𝒙) contains the point (3, 0)
(c) Sketch the graph of both 𝑓(𝑥) and 𝑓 !! (𝑥) on the same axes.
(d) How do the graphs compare to one another?
They reflect across the line 𝒚 = 𝒙, meaning that they are inverses of one another.
(e) For 𝑓(𝑥), give the domain, range, and the equation (and type) of the asymptote.
D: −∞, ∞
R: 𝟏, ∞
A: horizontal @ 𝒚 = 𝟏
!!
(f) For 𝑓 (𝑥), give the domain, range, and the equation (and type) of the asymptote.
D: 𝟏,∞
R: −∞, ∞
A: vertical @ 𝒙 = 𝟏
2. Look at an anonymous function represented by a table of values.
0 1 2 3 4
𝒙
𝒇(𝒙) 0 1 1 5 3
(a) Using the table below, give a table of values for the inverse of the above function.
0
𝒙
!𝟏
𝒇 (𝒙) 0
1
1
1
2
5
3
3
4
(b) Is the inverse a function? How can you tell?
No; the definition of a function requires that for every one 𝒙-value, there is only one 𝒚value. In this case, 𝒙 = 𝟏 has two different 𝒚-values.
3. Find the inverse of the following and give their domain.
!
(b) 𝑓 𝑥 = 𝑥 ! − 1
(a) 𝑔 𝑥 = !!!
𝟑
𝒈!𝟏 𝒙 = 𝒙 + 𝟏 and D: 𝒙 ≠ 𝟎
(c) 𝑓 𝑥 =
𝒇
!𝟏
!
𝒇!𝟏 𝒙 = ± 𝒙 + 𝟏 and D: 𝒙 ≥ −𝟏
!!!
!
𝟑
𝒙 = 𝟑𝒙 + 𝟕 and D: (−∞, ∞)
4. Verify: 𝒇!𝟏 𝒇 𝒙
= 𝒙 𝒂𝒏𝒅 𝒇 𝒇!𝟏 𝒙
(Show all steps!) = 𝒙 (a) Verify the function and the inverse you found in question 3 part (a) are inverses.
𝟑
𝟑
𝒙
𝒈 𝒈!𝟏 𝒙 =
= =𝟑∙ =𝒙
𝟑
𝟑
𝟑
𝒙+𝟏 −𝟏 𝒙
𝟑
𝒙−𝟏
𝒈!𝟏 𝒈 𝒙 =
+𝟏=𝟑∙
+𝟏=𝒙−𝟏+𝟏=𝒙
𝟑
𝟑
𝒙−𝟏
(a) Verify the function and the inverse you found in question 3 part (c) are inverses.
𝒇 𝒇!𝟏 𝒙
𝒇!𝟏 𝒇 𝒙
= 𝟑(
=
𝟑
𝟑
𝟑𝒙𝟑 + 𝟕 − 𝟕
=
𝟑
𝟑
𝟑𝒙𝟑
=
𝟑
𝟑
𝒙𝟑 = 𝒙
𝒙−𝟕 𝟑
𝒙−𝟕
) +𝟕=𝟑∙
+𝟕=𝒙−𝟕+𝟕=𝒙
𝟑
𝟑
To insert a graph into you answer document, take a screen shot or a picture. To show your work, write it out clearly by hand and take a picture or use the equation editor on your computer. Here is a link to an online graphing calculator: http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html To take a screen shot, press the print screen key (usually in the upper right corner of your key board) and paste into a Word document. You may also use a screen capture like Snipping Tool (already on many computers), Jing, Snagit, etc.