1. Verifying if two functions f,g are inverses of each other
f(x)=x+3. g(x)=x-3.
Solution:
Two functions f,g are inverses of each other only when 1. `f(g(x))=x` and 2. `g(f(x))=x`
1. Show that f(g(x))=x
`f(x)=x+3`
`g(x)=x-3`
`fog(x)=?`
`f(x)=x+3, g(x)=x-3, fog(x)=?`
`fog(x)=f(g(x))`
`=f(x-3)`
`=(x-3)+3`
`=x-3+3`
`=x`
`fog(x)=x`
2. Show that g(f(x))=x
`f(x)=x+3, g(x)=x-3, gof(x)=?`
`gof(x)=g(f(x))`
`=g(x+3)`
`=(x+3)-3`
`=x+3-3`
`=x`
`gof(x)=x`
Here both outputs are x, So f(x) and g(x) are inverses of each other
2. Verifying if two functions f,g are inverses of each other
f(x)=4x-3. g(x)=(x+3)/4.
Solution:
Two functions f,g are inverses of each other only when 1. `f(g(x))=x` and 2. `g(f(x))=x`
1. Show that f(g(x))=x
`f(x)=4x-3`
`g(x)=(x+3)/4`
`fog(x)=?`
`f(x)=4x-3, g(x)=(x+3)/4, fog(x)=?`
`fog(x)=f(g(x))`
`=f((x+3)/4)`
`=4*((x+3)/4)-3`
`=x+3-3`
`=x`
`fog(x)=x`
2. Show that g(f(x))=x
`f(x)=4x-3, g(x)=(x+3)/4, gof(x)=?`
`gof(x)=g(f(x))`
`=g(4x-3)`
`=((4x-3)+3)/4`
`=(4x)/4`
`=x`
`gof(x)=x`
Here both outputs are x, So f(x) and g(x) are inverses of each other
3. Verifying if two functions f,g are inverses of each other
f(x)=x/(x-1). g(x)=(2x)/(2x-1).
Solution:
Two functions f,g are inverses of each other only when 1. `f(g(x))=x` and 2. `g(f(x))=x`
1. Show that f(g(x))=x
`f(x)=x/(x-1)`
`g(x)=(2x)/(2x-1)`
`fog(x)=?`
`f(x)=x/(x-1), g(x)=(2x)/(2x-1), fog(x)=?`
`fog(x)=f(g(x))`
`=f((2x)/(2x-1))`
`=((2x)/(2x-1))/(((2x)/(2x-1))-1)`
`=((2x)/(2x-1))/((2x)/(2x-1)-1)`
`=(2x)/((2x-1)((2x)/(2x-1)-1))`
`=(2x)/((2x-1)((2x-(2x-1))/(2x-1)))`
`=(2x)/((2x-1)(1/(2x-1)))`
`=2x`
`fog(x)=2x`
The simplified answer is not `x`, So there is no need to do Step 2.
So we conclude f(x) and g(x) are not inverses of each other.
This material is intended as a summary. Use your textbook for detail explanation.
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