Problem 104 - College Algebra

Question

If equations for \(f\) and \(g\) are given, explain how to find \(f-g .\)

Step-by-Step Solution

Answer

The difference between function \(f\) and function \(g\), represented as \(f - g\), is obtained by subtracting the equation for \(g\) from that for \(f\). If the functions are \(f(x) = ax + b\) and \(g(x) = cx + d\), then \(f - g = (a-c)x + (b-d)\).

Step 1: Understand Function Subtraction

Firstly, you need to understand what subtracting functions means. If you have two functions, \(f(x)\) and \(g(x)\), their difference, denoted \(f(x)-g(x)\), simply involves subtracting each corresponding element in \(g(x)\) from \(f(x)\). Therefore, \(f(x) - g(x) = f(x) - g(x)\), where \(x\) is the element in the domain over which they are defined.

Step 2: Apply Function Subtraction

Once you understand function subtraction, the next step is to apply it on the given functions. If function \(f\) is represented as \(f(x) = ax + b\) and function \(g\) as \(g(x) = cx + d\), the difference \(f(x) - g(x)\) will be \(f(x) - g(x) = (ax + b) - (cx + d)\).

Step 3: Simplify the Expression

The last step is to simplify the resulting expression from step 2. This involves dealing with the brackets and combining like terms, if any. Using the distributive property of subtraction, the equation from step 2 simplifies to \(f(x) - g(x) = ax + b - cx - d = (a-c)x + (b-d)\). This is the expression for \(f-g\).