Key Concepts and Formulas

• Composition of Functions: For functions $f$ and $g$, the composite function $(f \circ g)(x)$ is defined as $f(g(x))$.
• Inverse Function Property: If $y = h^{-1}(x)$, then $h(y) = x$. This property is used to find the original function from its inverse.
• Equality of Polynomials: If two polynomials are equal for all values of the variable, then the coefficients of corresponding powers of the variable and the constant terms must be equal.
• Natural Numbers: The set of natural numbers $\mathbb{N}$ typically includes $\{1, 2, 3, \dots\}$.

Step-by-Step Solution

Step 1: Find the explicit form of $(f \circ g)(x)$ from its inverse.

We are given that $(f \circ g)^{-1}(x) = \left(\frac{x - 7}{2}\right)^{1/3}$. Let $y = (f \circ g)^{-1}(x)$. Then, by the property of inverse functions, $(f \circ g)(y) = x$. We have $y = \left(\frac{x - 7}{2}\right)^{1/3}$. To find $x$ in terms of $y$, we cube both sides: $y^3 = \frac{x - 7}{2}$ Multiply by 2: $2y^3 = x - 7$ Add 7 to both sides: $x = 2y^3 + 7$ Since $(f \circ g)(y) = x$, we have $(f \circ g)(y) = 2y^3 + 7$. Replacing the variable $y$ with $x$, we get the explicit form of the composite function: $(f \circ g)(x) = 2x^3 + 7$

Step 2: Express $(f \circ g)(x)$ using the given definitions of $f(x)$ and $g(x)$.

We are given $f(x) = ax - 3$ and $g(x) = x^b + c$, where $a, b, c \in \mathbb{N}$. The composition $(f \circ g)(x)$ is found by substituting $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x))$ Substitute $g(x) = x^b + c$ into $f(x)$: $(f \circ g)(x) = a(x^b + c) - 3$ Distributing $a$, we get: $(f \circ g)(x) = ax^b + ac - 3$

Step 3: Equate the two expressions for $(f \circ g)(x)$ to find the values of $a, b, c$.

From Step 1, we have $(f \circ g)(x) = 2x^3 + 7$. From Step 2, we have $(f \circ g)(x) = ax^b + ac - 3$. Equating these two expressions: $ax^b + ac - 3 = 2x^3 + 7$ For this equality to hold for all $x$, the coefficients of corresponding powers of $x$ and the constant terms must be equal.

Comparing the powers of $x$: The power of $x$ on the left is $b$, and on the right is $3$. Thus, $b = 3$.

Comparing the coefficients of $x^b$ (which is $x^3$ since $b=3$): The coefficient of $x^3$ on the left is $a$, and on the right is $2$. Thus, $a = 2$.

Comparing the constant terms: The constant term on the left is $ac - 3$, and on the right is $7$. Thus, $ac - 3 = 7$. Substitute the value of $a=2$ into this equation: $2c - 3 = 7$ $2c = 10$ $c = 5$ The values are $a=2$, $b=3$, and $c=5$. These are all natural numbers, satisfying the given condition $a, b, c \in \mathbb{N}$.

Step 4: Calculate $(f \circ g)(ac) + (g \circ f)(b)$.

First, determine the values of $ac$ and $b$: $ac = 2 \times 5 = 10$ $b = 3$

Now, we need to calculate $(f \circ g)(10)$ and $(g \circ f)(3)$.

Calculate $(f \circ g)(10)$: Using the expression from Step 1, $(f \circ g)(x) = 2x^3 + 7$: $(f \circ g)(10) = 2(10)^3 + 7 = 2(1000) + 7 = 2000 + 7 = 2007$

Calculate $(g \circ f)(3)$: First, find the expression for $(g \circ f)(x)$: $(g \circ f)(x) = g(f(x)) = g(ax - 3)$ Substitute $f(x) = 2x - 3$ into $g(x) = x^3 + 5$: $(g \circ f)(x) = (2x - 3)^3 + 5$ Now, substitute $x=3$: $(g \circ f)(3) = (2(3) - 3)^3 + 5$ $= (6 - 3)^3 + 5$ $= (3)^3 + 5$ $= 27 + 5 = 32$

Finally, add the two results: $(f \circ g)(ac) + (g \circ f)(b) = (f \circ g)(10) + (g \circ f)(3) = 2007 + 32 = 2039$

Common Mistakes & Tips

• Inverse Function Calculation: Ensure accurate algebraic manipulation when finding the inverse function. Remember that if $y = h^{-1}(x)$, then $h(y) = x$.
• Order of Composition: Be mindful of the order of functions in composition. $(f \circ g)(x)$ is $f(g(x))$, not $g(f(x))$.
• Equating Coefficients: When comparing polynomial expressions, ensure that you match coefficients of the same powers of $x$.

Summary

The problem requires finding the values of $a, b, c$ by equating two expressions for $(f \circ g)(x)$: one derived from its inverse and the other from the definitions of $f(x)$ and $g(x)$. Once $a, b, c$ are found, we calculate $(f \circ g)(ac)$ and $(g \circ f)(b)$ and sum them. The derived values $a=2, b=3, c=5$ satisfy the condition of being natural numbers. The final calculation yields $(f \circ g)(10) = 2007$ and $(g \circ f)(3) = 32$, resulting in a sum of $2039$.

The final answer is $\boxed{2039}$.