Problem 128 - College Algebra Essentials

Question

If \(f(x+y)=f(x)+f(y)\) and \(f(1)=3,\) find \(f(2), f(3)\) and \(f(4) .\) Is \(f(x+y)=f(x)+f(y)\) for all functions?

Step-by-Step Solution

Answer

The values of the function at the points are \(f(2) = 6\), \(f(3) = 9\) and \(f(4) = 12\). The equation \(f(x+y)=f(x)+f(y)\) is not true for all functions, as it characterizes linear functions only.

Step 1: Find \(f(2)\)

Substitute \(x = 1\) and \(y = 1\) in the equation \(f(x+y)=f(x)+f(y)\) to obtain \(f(1+1)=f(1)+f(1)\). Since \(f(1)=3\), we thus find that \(f(2) = 6\).

Step 2: Find \(f(3)\)

Now substitute \(x = 2\) and \(y = 1\) in the equation \(f(x+y)=f(x)+f(y)\) to obtain \(f(2+1)=f(2)+f(1)\). From Step 1 we know \(f(2)=6\) and from the problem we know \(f(1)=3\). By substituting these values we find \(f(3) = 9\).

Step 3: Find \(f(4)\)

Substitute \(x = 2\) and \(y = 2\) in the equation \(f(x+y)=f(x)+f(y)\) to obtain \(f(2+2)=f(2)+f(2)\). From Step 1, we know that \(f(2) = 6\). Substituting this we find \(f(4) = 12\).

Step 4: Analyze whether \(f(x+y)=f(x)+f(y)\) for all functions

The equation \(f(x+y)=f(x)+f(y)\) represents a particular type of mathematical function called a linear function. Not all functions are linear and many do not satisfy the property \(f(x+y)=f(x)+f(y)\). Simple counter-examples are quadratic or exponential functions. For example, if \(f(x)=x^2\), then \(f(x+y)=f(x)+f(y)\) is not satisfied for general \(x\) and \(y\).