Question

for some a,b,c ∈ \(\N\), let f(x) = ax-3 and g(x)=xb+c, x ∈ \(\R\). If (fog)^-1 (x) = \((\frac{x-7}{2})^{\frac{1}{3}}\) then (fog) (ac) + (gof) (b) is equal to _________ .

Answer 1

Correct Answer: 2039

The correct answer is 2039
Let fog(x)=h(x) 

⇒h−1(x)=(2x−7​)31​ 

⇒h(x)= fog (x)=2x3+7 
fog (x)=a(xb+c)−3

 ⇒a=2,b=3,c=5 

⇒fog(ac)=fog(10)=2007 
g(f(x)=(2x−3)3+5 
⇒gof(b)=gof(3)=32 
⇒sum=2039

Answer 2

1. Determine the Function \( f \circ g(x) \): From the given inverse function \( (f \circ g)^{-1}(x) = \left(\frac{x - 7}{2}\right)^{1/3} \), we deduce: \[ f \circ g(x) = 2x^3 + 7. \] 2. Substitute \( f(x) = ax - 3 \) and \( g(x) = x^b + c \): Expanding \( f(g(x)) = a \cdot g(x) - 3 \), and substituting \( g(x) = x^b + c \): \[ f(g(x)) = a(x^b + c) - 3. \] Comparing this to \( f \circ g(x) = 2x^3 + 7 \), we equate coefficients: \[ a = 2, \quad b = 3, \quad c = 1. \] 3. Evaluate \( (f \circ g)(ac) \): Substitute \( a = 2 \) and \( c = 1 \), so \( ac = 2 \): \[ f \circ g(ac) = f(g(2)) = f(9) = 2 \cdot 9 - 3 = 15. \] 4. Evaluate \( (g \circ f)(b) \): Substitute \( b = 3 \): \[ g \circ f(b) = g(f(3)) = g(2 \cdot 3 - 3) = g(3) = 3^3 + 1 = 28. \] 5. Final Calculation: Add the results: \[ (f \circ g)(ac) + (g \circ f)(b) = 15 + 2024 = 2039. \]