For some a, b, c N, let f ( x ) = a x - 3 and g ( x ) = x b + c , x ? R . If ( f o g ) - 1 ( x ) = ( x - 7 2 ) 1 / 3 , then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

Ans.
(2039)

Given, $f (x) = a x - 3, g (x) = x^{b} + c$ and ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{\frac{1}{3}}$

Let $f o g (x) = h^{- 1} (x) .$ So, $h^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3}$

Let $y = {(\frac{x - 7}{2})}^{1 / 3}$ $\Rightarrow$ $y^{3} = \frac{x - 7}{2} \Rightarrow x = 2 y^{3} + 7$

$∴$ $h (x) = f o g (x) = 2 x^{3} + 7$

Now, $f o g (x) = a (x^{b} + c) - 3 \Rightarrow 2 x^{3} + 7 = a (x^{b} + c) - 3$

Comparing the coefficients of like powers, we get

$a = 2, b = 3$ and $c = 5$

So, $f o g (a c) = f o g (10) = 2 {(10)}^{3} + 7 = 2007$

$g [f (x)] = {(a x - 3)}^{b} + c = {(2 x - 3)}^{3} + 5$

$g o f (b) = (2 {(3) - 3)}^{3} + 5 = 27 + 5 = 32$

$∴ f o g (a c) + g o f (b) = 2007 + 32 = 2039$

Want Us to Email you About Special Offers & Updates?