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Composition of Functions:
Inverse Functions and Composition (page 6 of 6)
104 2x 2x 50 2x. 106 2 3 4 1 1x1x 1x 1x 1x 1x. 107 5 6 1x 2x 7 1x 1x 8 2x. 108 9 2x 1x 10 2x 1x 11 1x 2x 2x. 109 12 1x 13 4x 2x 4x 2x 4x 4x 1 2 2x.
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
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:
- 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):
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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)
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)
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
Inverting the composed function:
( fog)(x) = x + 7
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)
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)
PDF X is the free PDF reader & editor 2020 for windows, an alternative to adobe acrobat reader. It's a powerful app for viewing, printing, editing and annotating PDFs. Features Supported Formats: PDF, PS, Tiff, CHM, DjVu, Images, DVI, XPS, ODT, Fiction Book, Comic Book, Plucker, EPub, Fax VIEW. Horizontal or Vertical scroll, Single Page. NC MATH 1—RELEASED ITEMS 3 Go to the next page. 3 Which expression is equivalent to (x + 2)(3x – 3)?A 3x2 – 6 B 3x2 + 3x – 6 C 3x2 + 6x – 6 D 3x2 + 9x – 6 RELEASED. In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. PDF-XChange Editor PDF-XChange Editor Plus. 1997-2021 Tracker Software Products - A wholly owned subsidiary of PDF-XChange Co Ltd. Registered in England: N0.
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.
Pdf Plus 1 3 X 2
- 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)
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)
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
Inverting the composed function:
( fog)(x) = x + 7
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)
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)
PDF X is the free PDF reader & editor 2020 for windows, an alternative to adobe acrobat reader. It's a powerful app for viewing, printing, editing and annotating PDFs. Features Supported Formats: PDF, PS, Tiff, CHM, DjVu, Images, DVI, XPS, ODT, Fiction Book, Comic Book, Plucker, EPub, Fax VIEW. Horizontal or Vertical scroll, Single Page. NC MATH 1—RELEASED ITEMS 3 Go to the next page. 3 Which expression is equivalent to (x + 2)(3x – 3)?A 3x2 – 6 B 3x2 + 3x – 6 C 3x2 + 6x – 6 D 3x2 + 9x – 6 RELEASED. In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. PDF-XChange Editor PDF-XChange Editor Plus. 1997-2021 Tracker Software Products - A wholly owned subsidiary of PDF-XChange Co Ltd. Registered in England: N0.
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.
Pdf Plus 1 3 X 2
Lightwave 3d 2018 0 3 – 3d animation software. << PreviousTop |1 | 2 | 3 | 4 | 5 | 6 | Return to Index
Pdf Plus 1 3 X 2/3
Cite this article as: | Stapel, Elizabeth. 'Inverse Functions and Composition.' Purplemath. Available from |
