GATE Overflow Test Series | Mixed Subjects | Test 1 | Question: 23

Question

If $f(x)=5x+2$ and $g(x)=3x-4$, what is the value of $a$ if $(f\circ g^{-1}\circ f^{-1})(a) = 4?$

Answer 1

Given that $f(x)=5x+2$ and $g(x)=3x-4$

Inverse of the function $: f^{-1}(x)=\dfrac{x-2}{5}$ and $: g^{-1}(x)=\dfrac{x+4}{3}$

Now, $(f\circ g^{-1}\circ  f^{-1})(a) = f(g^{-1}(f^{-1}(a)))=f\left(g^{-1}\left(\dfrac{a-2}{5}\right)\right)$

$= f\left(\dfrac{\dfrac{a-2}{5} + 4}{3}\right) = f\left(\dfrac{a+18}{15}\right)$

$= 5 \times \left(\dfrac{a+18}{15}\right) + 2 = \dfrac{a+24}{3}$

$ \dfrac{a+24}{3} = 4$

$\implies a=-12$

So, the correct answer is $-12$.