Question

find range of the functions 1)if f (x)=ax+b ÷cx-a ,then find fof (x)

Answer 1

I find range as well as a fof in the attached picture

Answer 2

Given,
$f(x)= \frac{ax+b}{cx-a}$

As asked for, let's find out the range for all x ∈ domain of $f(x)$
$y=\frac{ax+b}{cx-a}$
⇒ $y(cx-a) = ax+b$
⇒ $cxy - ay = ax + b$
Let's bring x together.
⇒ $cxy - ax = b + ay$
⇒ $x(cy-a) = b + ay$
⇒ $x= \frac{b+ay}{cy-a}$
For range of x, cy-a≠0
⇒ y≠    $\frac{a}{c}$

Hence, y ∈ R - $\left[\begin{array}{ccc} \frac{a}{c} \end{array}\right]$ ∨ x ∈ domain.

Also, fof(x) shall be found by replacing x in f(x) with f(x).
⇒ $fof(X) = \frac{a\frac{ax+b}{cx-a}+b}{c\frac{ax+b}{cx-a}-a}$
⇒ $fof(x) = \frac{a^{2}x + ab + bcx - ab }{acx + bc - acx + a^{2} }$
⇒$fof(x) = ( \frac{ a^{2} + bc }{a^{2} + bc} )x$
⇒fof(x) = x.

Hence, fof(x) = x.

Solved.