"Let f : R $-$ {3} $\to$ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.Let g : R $\to$ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f$-$1(x) + g$-$1(x) = ${{13} \over 2}$ is equal to :"

Question

Accepted Answer

"Finding inverse of f(x)

$$y = {{x - 2} \over {x - 3}} \Rightarrow xy - 3y = x - 2 \Rightarrow x(y - 1) = 3y - 2$$

$\therefore$ ${f^{ - 1}}(x) = {{3x - 2} \over {x - 1}}$

Similarly for ${g^{ - 1}}(x)$

$$y = 2x - 3 \Rightarrow x = {{y + 3} \over 2} \Rightarrow {g^{ - 1}}(x) = {{x + 3} \over 2}$$

$\therefore$ ${{3x - 2} \over {x - 1}} + {{x + 3} \over 2} = {{13} \over 2}$

$\Rightarrow 6x - 4 + {x^2} + 2x - 3 = 13x - 13$

$\Rightarrow {x^2} - 5x + 6 = 0$

$\Rightarrow (x - 2)(x - 3) = 0$

$\Rightarrow$ x = 2 or 3"

Let f : R $-$ {3} $\to$ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.

Let g : R $\to$ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f^$-$1(x) + g^$-$1(x) = ${{13} \over 2}$ is equal to :

Solution

About this question

Drill 25 more like these. Every day. Free.

Let f : R $-$ {3} $\to$ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.Let g : R $\to$ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f$-$1(x) + g$-$1(x) = ${{13} \over 2}$ is equal to :

Solution

About this question

More Sets, Relations and Functions questions

Drill 25 more like these. Every day. Free.

Let f : R $-$ {3} $\to$ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.

Let g : R $\to$ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f^$-$1(x) + g^$-$1(x) = ${{13} \over 2}$ is equal to :