Question

Let \(f:\R→\R\) be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ \(\R\), and \(g:\R→(0,\infin)\) be a function such that g(x + y) = g(x)g(y) for all x, y ∈ \(\R\). If \(f(\frac{-3}{5})=12\) and \(g(\frac{-1}{3})=2\), then the value of \((f(\frac{1}{4})+g(-2)-8)g(0)\) is ______.

Answer 1

Correct Answer: 51

Functional Equations 

The functional equations imply:

Given:

• f(x + y) = f(x) + f(y) ⇒ f(x) = kx
• g(x + y) = g(x)g(y) ⇒ g(x) = ax

Step 1: Solve for k and a

From the equation \( f(-3) = 12 \), we get:

\[ k \cdot \left( -\frac{3}{5} \right) = 12 \quad \Rightarrow \quad k = -20 \]

Similarly:

Substituting \( g\left( -\frac{1}{3} \right) = 2 \), we get:

\[ a \cdot \left( -\frac{1}{3} \right) = 2 \quad \Rightarrow \quad a = \frac{1}{8} \]

Step 2: Calculate function values

• \(f\left(\frac{1}{4}\right) = -20 \frac{1}{4} = -5\)
• \(g(-2) = \left( \frac{1}{8} \right)^{-2} = 64\)
• g(0) = 1

Step 3: Final Calculation

Now, calculate:

\[ f\left( \frac{1}{4} \right) + g(-2) - 8 \cdot g(0) = -5 + 64 - 8 \cdot 1 = 51 \]