Animated Solution for Mathematics - Functions: For some $a,b,c\in \mathbb{N}$, let $f(x)=a{x}^b+3$ and $g(x)=x^b+c,x\in \mathbb{R}$. If $(f\circ g{)}^{-1}(x)={\left(\frac{x-7}{2}\right)}^{1/3}$, then $(f\circ g)(ac)+(g\circ f)(b)$ is equal to

Welcome, fellow explorer of the mathematical universe! Today, we are going to peel back the layers of a classic function composition problem. It might look like a dense thicket of variables and exponents, but I promise you, once we find the right path, the logic flows as beautifully as a symphony.

The Power of Reversal

Imagine you have a machine that takes an input $x$ and transforms it into an output. An inverse function, denoted as $f^{-1}(x)$, is simply that machine running in reverse. We are given the inverse of a composite function:

To find the original function $(f\circ g)(x)$, we need to 'undo' the inverse. Let $y={\left(\frac{x-7}{2}\right)}^{1/3}$. If we cube both sides, we get:

Now, multiply by $2$ to get $2y^3=x-7$, and finally, $x=2y^3+7$. By swapping the variables back, we have successfully reconstructed our original composite function:

The Algebraic Bridge

Now, let us look at the definitions provided: $f(x)=ax-3$ and $g(x)=x^b+c$. When we compose these, we are essentially feeding the output of $g$ into $f$. So:

Expanding this, we get $a{x}^b+ac-3$. We now stand at a fascinating crossroads. We have two different expressions for the exact same function $(f\circ g)(x)$: one from our inverse analysis ($2x^3+7$) and one from our algebraic expansion ($a{x}^b+ac-3$).

By the principle of identity, these must be equal for all $x$:

Comparing the variable terms, it becomes crystal clear that $b=3$ and $a=2$. With $a=2$, we can easily solve for $c$ by comparing the constants:

The Final Calculation

With $a=2$, $b=3$, and $c=5$, our functions are $f(x)=2x-3$ and $g(x)=x^3+5$. Now, we simply evaluate the final expression: $(f\circ g)(ac)+(g\circ f)(b)$.

First, $ac=2\times 5=10$. We need $(f\circ g)(10)$. Using our formula $(f\circ g)(x)=2x^3+7$, we get:

Next, we need $(g\circ f)(b)$, which is $(g\circ f)(3)$. Working from the inside out, $f(3)=2(3)-3=3$. Then:

Adding these together, $2007+32=2039$. And there it is—the final answer, 2039. It is not just a number; it is the culmination of our systematic, step-by-step reasoning.