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]

Question

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]

Smart Achievers · Accepted Answer

(2039)

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

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

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

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

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]

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Solution

Q. For some a, b, c∈N, let f(x)=ax-3 and g(x)=xb+c,x∈R. If (fog)-1(x)=(x-72)1/3, then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

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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]