Question

If 5f ( x+y ) = f(x).f(y) and f(3) = 320, then the value of f(1) is

Answer 1

Correct Answer: 20

Given the functional equation \(5f(x+y) = f(x) \cdot f(y)\) and \(f(3) = 320\), we aim to find \(f(1)\).

Finding f(0)

Set \(x = 0\) and \(y = 0\):

\(5f(0+0) = f(0) \cdot f(0)\)

\(5f(0) = f(0)^2\)

Rearrange and factor:

\(f(0)^2 - 5f(0) = 0\)

\(f(0)(f(0) - 5) = 0\)

Thus, \(f(0) = 0\) or \(f(0) = 5\).

Rejecting f(0) = 0

If \(f(0) = 0\), then using \(y=0\) in the functional equation would give \(5f(x) = f(x) * 0\), thus \(f(x)=0\) for all values of x. This contradicts that \(f(3) = 320\). Therefore, \(f(0) = 5\).

Expressing f(3) in terms of f(1)

Using the functional equation:

\(5f(1+1) = f(1) \cdot f(1)\) => \(5f(2) = f(1)^2\) => \(f(2) = \frac{f(1)^2}{5}\)

Similarly:

\(5f(2+1) = f(2) \cdot f(1)\) => \(5f(3) = \frac{f(1)^2}{5} \cdot f(1)\) => \(5f(3) = \frac{f(1)^3}{5}\) => \(f(3) = \frac{f(1)^3}{25}\)

Using f(3) = 320

Given \(f(3) = 320\), substitute this into the above equation:

\(320 = \frac{f(1)^3}{25}\)

\(320 * 25 = f(1)^3\)

\(8000 = f(1)^3\)

Take the cube root of both sides:

\(f(1) = \sqrt[3]{8000} = 20\)

General Form of f(x)

Based on the provided initial steps and corrected steps, the general form for the function f(x) is \(f(x) = 5 \cdot 4^x\)

where \(f(0) = 5\) , \(f(1) = 20\), \(f(2) = 80\) , \(f(3) = 320\)

The value of \(f(1)\) is 20.