![Soulver Soulver](https://pixellibre.net/streisand-data/autoblog/Korben/index.php?m=https://korben.info/app/uploads/2019/12/IMG_6401-473x1024.png)
Download Soulver 2 for macOS 10.10.0 or later and enjoy it on your Mac. Soulver helps you do quick calculations and work things out. Use Soulver to play around with numbers, do 'back of the envelope' calculations, and solve day-to-day problems. View 3x^3+4x^2-7x+2=0.png from MATH 109 at Georgia Military College Valdosta Campus. ID: 3 x 3 + 4 x- 7 x + 2= 0 Step 1: Write ux' as a sum and - 7% of a difference.
Return to the Lessons Index | Do the Lessons in Order | Print-friendly page |
Composition of Functions:
Inverse Functions and Composition (page 6 of 6)
Inverse Functions and Composition (page 6 of 6)
Soulver 3 0 4 X 20
Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition
Soulver 3
The lesson on inverse functions explains how to use function composition to verify that two functions are inverses of each other. However, there is another connection between composition and inversion:
Soulver 3 0 4 X 200
- Given f (x) = 2x – 1 and
g(x) = (1/2)x + 4,
find f–1(x), g–1(x), ( fog)–1(x),
and (g–1of –1)(x).
What can you conclude?
This involves a lot of steps, so I'll stop talking and just show you how it goes.
First, I need to find f–1(x), g–1(x), and ( fog)–1(x):
Advertisement
Inverting f (x):
f (x) = 2x – 1
y = 2x – 1
y + 1 = 2x
(y + 1)/2 = x
(x + 1)/2 = y
(x + 1)/2 = f–1(x)
y = 2x – 1
y + 1 = 2x
(y + 1)/2 = x
(x + 1)/2 = y
(x + 1)/2 = f–1(x)
Inverting g(x):
g(x) = (1/2)x + 4
y = (1/2)x + 4
y – 4 = (1/2)x
2(y – 4) = x
2y – 8 = x
2x – 8 = y
2x – 8 = g–1(x)
y = (1/2)x + 4
y – 4 = (1/2)x
2(y – 4) = x
2y – 8 = x
2x – 8 = y
2x – 8 = g–1(x)
Finding the composed function: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
( fog)(x) = f (g(x)) = f ((1/2)x + 4)
= 2((1/2)x + 4) – 1
= x + 8 – 1
= x + 7
= 2((1/2)x + 4) – 1
= x + 8 – 1
= x + 7
Inverting the composed function:
Boom2:volume boost & equalizer 1 6 2. ( fog)(x) = x + 7
y = x + 7
y – 7 = x
x – 7 = y
x – 7 = ( fog)–1(x)
y = x + 7
y – 7 = x
x – 7 = y
x – 7 = ( fog)–1(x)
Now I'll compose the inverses of f(x) and g(x) to find the formula for (g–1of –1)(x):
(g–1of –1)(x) = g–1( f–1(x))
= g–1( (x + 1)/2 )
= 2( (x + 1)/2 ) – 8
= (x + 1) – 8
= x – 7 = (g–1of –1)(x)
= g–1( (x + 1)/2 )
= 2( (x + 1)/2 ) – 8
= (x + 1) – 8
= x – 7 = (g–1of –1)(x)
Note that the inverse of the composition (( fog)–1(x)) gives the same result as does the composition of the inverses ((g–1of –1)(x)). So I would conclude that
( fog)–1(x) = (g–1of –1)(x)
While it is beyond the scope of this lesson to prove the above equality, I can tell you that this equality is indeed always true, assuming that the inverses and compositions exist — that is, assuming there aren't any problems with the domains and ranges and such.
<< PreviousTop |1 | 2 | 3 | 4 | 5 | 6 | Return to Index
Cite this article as: | Yummy ftp watcher 3 0 8. Stapel, Elizabeth. 'Inverse Functions and Composition.' Purplemath. Available from https://www.purplemath.com/modules/fcncomp6.htm. Accessed [Date] [Month] 2016 |