Inverse Functions Homework: Find f^-1, g^-1, Show f^-1 f=x

[kingstar]

Homework Statement

5. (a) The functions f and g are defined by
f : x|→ 2x + ln3 (x is a real number)
g : x|→ e^3x (x is a real number)

i) Find f^-1(x) and g^-1(x) and state their domain of definition

ii) Show that f^-1 f = f f^-1 = x (x is a real number)

iii) Find the composite function gf x( )and state the range.

Homework Equations

The Attempt at a Solution

So i worked out the inverse for each.

f^-1(x) = 1/2(x-ln3)
g^-1(x) = 1/3(lnx)

But for part two I'm not sure what this means, does it mean that i should sub my inverse function into my original function and then vice versa?

Also when i double checked my answer, it has log's in place of my ln's. Was i meant to use log instead of ln? :S

 

[Dick]

Homework Statement

5. (a) The functions f and g are defined by
f : x|→ 2x + ln3 (x is a real number)
g : x|→ e^3x (x is a real number)

i) Find f^-1(x) and g^-1(x) and state their domain of definition

ii) Show that f^-1 f = f f^-1 = x (x is a real number)

iii) Find the composite function gf x( )and state the range.

Homework Equations

The Attempt at a Solution

So i worked out the inverse for each.

f^-1(x) = 1/2(x-ln3)
g^-1(x) = 1/3(lnx)

But for part two I'm not sure what this means, does it mean that i should sub my inverse function into my original function and then vice versa?

Also when i double checked my answer, it has log's in place of my ln's. Was i meant to use log instead of ln? :S

Yes. That's exactly what it means. You should get x for both answers, that's the identity function. And here 'log' and 'ln' mean the same thing. 'log' can be used if you are talking about any base. If you want to be specific then you need to indicate the base, i.e. $log_2$. 'ln' is $log_e$.

 

[kingstar]

Thanks.