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Determine The Inverse G(X) Of The Function F(X) = (1+(4/X)), Stating Its Domain And Range. Verify That F(G(X)) = G(F(X))=X And That G′(F(X)) = (1/(F′(X))
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Calculus > inverse function
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Peter - this was the 'Car polishing' thread.
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Hi PP.
Yes. Start by writing f(x) as y=1+(4/x)
So y-1=4/x
so x = 4/(y-1)
so to find the inverse g(x) change x to y and y to x. *
So g(x) is 'y' =4/(x-1)
You can check it by putting in some numbers. eg try x= 4
f(x)= 1+(4/4)=2. Think of that as the output.
Use that output as the input into the inverse, g(x) and we get g(x)=4/(2-1) =4, so gets us back to where we started
Now try x=0.5
f(x)= 1+(4/0.5)= 9
g(x)=4/(9-1)= 0.5.
Ta da!
Works for any initial input value (except x=0 since can't divide 4 by 0 )
*I know jim probably wouldn't approve of this GCSE type method for 'doing' these whereby you swap x and y round but it gets to the answer.
Anyway- must leave it there as I've been furloughed and am not allowed to do this stuff.
Yes. Start by writing f(x) as y=1+(4/x)
So y-1=4/x
so x = 4/(y-1)
so to find the inverse g(x) change x to y and y to x. *
So g(x) is 'y' =4/(x-1)
You can check it by putting in some numbers. eg try x= 4
f(x)= 1+(4/4)=2. Think of that as the output.
Use that output as the input into the inverse, g(x) and we get g(x)=4/(2-1) =4, so gets us back to where we started
Now try x=0.5
f(x)= 1+(4/0.5)= 9
g(x)=4/(9-1)= 0.5.
Ta da!
Works for any initial input value (except x=0 since can't divide 4 by 0 )
*I know jim probably wouldn't approve of this GCSE type method for 'doing' these whereby you swap x and y round but it gets to the answer.
Anyway- must leave it there as I've been furloughed and am not allowed to do this stuff.
Okay, if anyone else is trying to refresh their memory on these:
Now we know g(x) and f(x), we can do step 3 which is to find g(f(x)) and f (g(x)). We can do this using algebraic substitution
To find g (f(x)) you apply the g(x) function to f(x).
So take g(x), which is 4/(x-1) and replace the x by f(x) which we were told is 1+(4/x) and we get our answer of....x.
Rinse and repeat for f(g(x)
Now we know g(x) and f(x), we can do step 3 which is to find g(f(x)) and f (g(x)). We can do this using algebraic substitution
To find g (f(x)) you apply the g(x) function to f(x).
So take g(x), which is 4/(x-1) and replace the x by f(x) which we were told is 1+(4/x) and we get our answer of....x.
Rinse and repeat for f(g(x)
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