MathematicsFor some a, b, c∈N, let f(x)=ax−3 and g(x)=xb+c, x∈R. If fog−1x=x−7213 then (fog) (ac) + (gof) (b) is equal to …….

For some $a, b, c \in N, let f (x) = ax - 3 and g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{\frac{1}{3}}$ then $(f o g) (a c) + (g o f) (b)$ is equal to …….

Solution:

$f (x) = ax - 3 and g (x) = x^{b} + c, x \in R .$
$f (g (x)) = a (x^{b} + c) - 3 = {ax}^{b} + ac - 3 ∴ {(f (g (x)))}^{- 1} = {(\frac{x + 3 - ac}{a})}^{\frac{1}{b}} = = {(\frac{x - 7}{2})}^{\frac{1}{3}} ∴ b = 3; a = 2; c = 5$
$f (x) = 2 x - 3, g (x) = x^{3} + 5$
$(f o g) (a c) + (g o f) (b) = (f o g) (10) + (g o f) (3) = f (1005) + g (3) = 2007 + 32 = 2039$

For some $a, b, c \in N, let f (x) = ax - 3 and g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{\frac{1}{3}}$ then $(f o g) (a c) + (g o f) (b)$ is equal to …….

Solution:

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