Question

4. If f(x)= x/x-1 ,x ne1
(a) Find fof * (x)
(b) Find the inverse of f

Answer 1

Answer:

f(x)=x−1x+1

fof(x)=f(x)−1f(x)+1=x−1x+1−1x−1x+1+1=x+1−x+1x+1+x−1=x

i)(fofof)(x)=f(fof(x))=f(x)=x−1x+1

ii)(fofofof)(x)=fof(fof(x))=fof(x)=x

Step-by-step explanation:

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Answer 2

Concept:

This question requires

$fof(x)=f(f(x))$

To find the inverse of a function, write the characteristic y as a function of x i.e. Y = f(x) after which solve for x as a characteristic of y.

Given:

The following equation is given,

$f(x)=\frac{x}{x-1}$,  x∈1

To find:

We need to find

(a) Find $fof(x)$

(b) Find $f^{-1}(x)$

Solution:

(a) We know that

$f(x)=\frac{x}{x-1}$

Therefore,

$fof(x)=f(f(x))$

⇒$fof(x)=f(\frac{x}{x-1} )$

⇒$fof(x)=\frac{\frac{x}{x-1} }{\frac{x}{x-1} -1}$

⇒$fof(x)=\frac{\frac{x}{x-1} }{\frac{x-x+1}{x-1} }$

⇒$fof(x)=\frac{\frac{x}{x-1} }{\frac{1}{x-1} }$

⇒$fof(x)=x$

Therefore the value of $fof(x)$ is x.

(b) Let g(x) be the inverse of f(x)

Let, f(x)=y=$\frac{x}{x-1}$

⇒y(x-1) = x

⇒xy-y = x

⇒ xy-x = y

⇒x(y-1) = y

⇒ x = $\frac{y}{y-1}$

Therefore,

if f(x) = y

Then,

g(y) = x

⇒g(y) = $\frac{y}{y-1}$

⇒ g(x) = $\frac{x}{x-1}$

Therefore, $f^{-1}(x)$ is $\frac{x}{x-1}$.