For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-6

Verified step by step guidance

1. Identify the given functions: \( f(x) = 3x + 4 \) and \( g(x) = 2x - 6 \).
2. Find the difference of the functions, which is defined as \( (f - g)(x) = f(x) - g(x) \).
3. Substitute the expressions for \( f(x) \) and \( g(x) \) into the difference: \( (f - g)(x) = (3x + 4) - (2x - 6) \).
4. Simplify the expression by distributing the negative sign and combining like terms: \( (f - g)(x) = 3x + 4 - 2x + 6 \).
5. Determine the domain of each function. Since both \( f(x) = 3x + 4 \) and \( g(x) = 2x - 6 \) are linear functions, their domains are all real numbers, \( (-\infty, \infty) \). The domain of \( (f - g)(x) \) is also all real numbers.

Key Concepts

Function Operations (Addition and Subtraction)

Function operations involve combining two functions by adding or subtracting their outputs for the same input value. For (ƒ - g)(x), subtract g(x) from ƒ(x) to create a new function. This helps analyze how two functions interact algebraically.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all operations are valid.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are defined for all real numbers. Understanding their properties simplifies finding combined functions and their domains.