Inverse Function question (1 Viewer)

[StudyHard123]

The function f(x) is defined by:

f(x) = ln[ 1/(1+x) ] , -1 < x <= 1

(i) Explain why f(x) has an inverse.

(ii) Find the domain of the inverse function.

Thanks

 

[pikachu975]

The function f(x) is defined by:

f(x) = ln[ 1/(1+x) ] , -1 < x <= 1

(i) Explain why f(x) has an inverse.

(ii) Find the domain of the inverse function.

Thanks

i) f(x) = ln1 - ln(1+x)
f(x) = -ln(1+x)
Derivative:
f'(x) = -1/(1+x)
Since the domain is -1 < x <= 1, then 0 < x+1 <= 2
So f'(x) is always negative, hence f(x) is monotonic decreasing so there is an inverse.

ii) The domain of the inverse is the range of the main function
Range of f(x): As x approaches -1, 1/(1+x) approaches infinity, so ln(1/(1+x)) approaches infinity.
For x = 1, f(x) = ln(1/2) = -ln2
Therefore range: y>= -ln2

Therefore domain of inverse is x >= - ln2

 

[si2136]

The function has an inverse because it is a one to one function. Use Desmos if you want to see what it looks like.

Find the inverse function and find where the domain lies. One approaches 0 - and the other approaches infinity+

GL

 

[pikachu975]

The function has an inverse because it is a one to one function. Use Desmos if you want to see what it looks like.

Find the inverse function and find where the domain lies. One approaches 0 - and the other approaches infinity+

GL

Depending on how many marks, part i might require the derivative to prove it if it was say 2 marks. I think if it was 1 mark it would just be stating one to one. Also there's a restriction on the domain of the original function which would affect the range of the inverse.