Find f−g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

Verified step by step guidance

1. Identify the given functions: \(f(x) = 2x + 3\) and \(g(x) = x - 1\).
2. Find the difference of the functions, which is \(f - g\). This means subtract \(g(x)\) from \(f(x)\): write the expression \(f(x) - g(x)\).
3. Substitute the given functions into the expression: \(f(x) - g(x) = (2x + 3) - (x - 1)\).
4. Simplify the expression by distributing the negative sign and combining like terms: \$2x + 3 - x + 1$.
5. Determine the domain of \(f - g\). Since both \(f(x)\) and \(g(x)\) are linear functions defined for all real numbers, the domain of \(f - g\) is all real numbers.

Key Concepts

Function Subtraction (f - g)

Function subtraction involves creating a new function by subtracting the output of one function from another for the same input value. For functions f(x) and g(x), (f - g)(x) = f(x) - g(x). This operation combines the two functions into one, which can then be analyzed or simplified.

Domain of a Function

The domain of a function is the set of all input values (x-values) 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 for those inputs.

Linear Functions

Linear functions are polynomial functions of degree one, typically written as f(x) = mx + b, where m and b are constants. They have a constant rate of change and their graphs are straight lines. Understanding their properties helps in performing operations like addition or subtraction and determining domains.