Verifying Inverse Functions Using Composition

[kennabah]

Verify using composition that F(x)=1/(x-1) and g(x)= (1+x)/x are inverse functions of each other.

I know how to verify inverse functions pretty easily the only problem i have is that the two x's in g(x) are throwing me off. I don't know what to do with them and without that i don't know how to start.

 

[royhaas]

Solve \(\displaystyle (1+x)/x = y\) for \(\displaystyle x\).

 

[Deleted member 4993]

Verify using composition that F(x)=1/(x-1) and g(x)= (1+x)/x are inverse functions of each other.

I know how to verify inverse functions pretty easily the only problem i have is that the two x's in g(x) are throwing me off. I don't know what to do with them and without that i don't know how to start.

You need to write f(g(x)

\(\displaystyle f(g(x)) \, = \, \frac{1}{1 \, - \, g(x)} \, = \, \frac{1}{1 \, - \, \frac{1+x}{x}}\)

Now continue....

 

[kennabah]

Would this be correct?

f(g(x))= 1 / ((1+x/x)-1)
= 1 / ((1+x-x)/x)
= 1 / (1/x)
=x
f(g(x))=x


g(f(x))= (1+(1/(x-1)) / (1/x-1)
= (x-1+1/(x-1) / (1/(x-1))
= (x/(x-1)) / (1/(x-1))
= x/(x-1) multiplied by (x-1)/1
the two (x-1)'s cross out and leave
g(f(x)) to = x

f(g(x))=g(f(x))=x

 

[Deleted member 4993]

Would this be correct?

f(g(x))= 1 / ((1+x/x)-1)
= 1 / ((1+x-x)/x)
= 1 / (1/x)
=x
f(g(x))=x


g(f(x))= (1+(1/(x-1)) / (1/x-1)
= (x-1+1/(x-1) / (1/(x-1))
= (x/(x-1)) / (1/(x-1))
= x/(x-1) multiplied by (x-1)/1
the two (x-1)'s cross out and leave
g(f(x)) to = x

f(g(x))=g(f(x))=x

Looks like you got it.....