Value of Combined Expression of Additive and Multiplicative Functional Equations

To find the value of the given expression, we analyze the properties of the functional equations for $f(x)$ and $g(x)$.

Step 1: Evaluation of $f\left(\frac{1}{4}\right)$

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies Cauchy's additive functional equation: $f(x + y) = f(x) + f(y) \quad \text{for all } x, y \in \mathbb{R}$

By mathematical induction, for any rational number $r \in \mathbb{Q}$ and any $x \in \mathbb{R}$, we have: $f(r x) = r f(x)$

We are given $f\left(-\frac{3}{5}\right) = 12$. Expressing $\frac{1}{4}$ in terms of $-\frac{3}{5}$, we have: $\frac{1}{4} = \left(-\frac{5}{12}\right) \times \left(-\frac{3}{5}\right)$

Since $-\frac{5}{12}$ is a rational number, we apply the property: $f\left(\frac{1}{4}\right) = f\left(-\frac{5}{12} \cdot \left(-\frac{3}{5}\right)\right) = -\frac{5}{12} f\left(-\frac{3}{5}\right)$

Substitute $f\left(-\frac{3}{5}\right) = 12$: $f\left(\frac{1}{4}\right) = -\frac{5}{12} \times 12 = -5$

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Step 2: Evaluation of $g(0)$ and $g(-2)$

The function $g: \mathbb{R} \rightarrow (0, \infty)$ satisfies the exponential functional equation: $g(x + y) = g(x) g(y) \quad \text{for all } x, y \in \mathbb{R}$

1. Finding $g(0)$: Setting $x = y = 0$: $g(0) = g(0 + 0) = g(0) g(0) = (g(0))^2$ Since the codomain of $g$ is $(0, \infty)$, $g(0) \neq 0$. Dividing both sides by $g(0)$, we get: $g(0) = 1$

2. Finding $g(-2)$: For any rational number $r \in \mathbb{Q}$ and any $x \in \mathbb{R}$, the relation $g(r x) = (g(x))^r$ holds.

   Expressing $-2$ in terms of $-\frac{1}{3}$, we have: $-2 = 6 \times \left(-\frac{1}{3}\right)$

   Therefore: $g(-2) = g\left(6 \cdot \left(-\frac{1}{3}\right)\right) = \left(g\left(-\frac{1}{3}\right)\right)^6$

   Given $g\left(-\frac{1}{3}\right) = 2$: $g(-2) = 2^6 = 64$

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Step 3: Calculating the Final Expression

We substitute $f\left(\frac{1}{4}\right) = -5$, $g(-2) = 64$, and $g(0) = 1$ into the target expression: $\left( f\left(\frac{1}{4}\right) + g(-2) - 8 \right) g(0) = (-5 + 64 - 8) \times 1$

$= (59 - 8) \times 1 = 51$

Thus, the value of the expression is 51.