In Exercises 31–50, find f−g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the expression for \( (f-g)(x) \), which is the difference between the functions \( f(x) \) and \( g(x) \).
2. Step 2: Write down the expressions for \( f(x) \) and \( g(x) \). We have \( f(x) = x - 5 \) and \( g(x) = 3x^2 \).
3. Step 3: Calculate \( (f-g)(x) \) by subtracting \( g(x) \) from \( f(x) \). This gives us \( (f-g)(x) = f(x) - g(x) = (x - 5) - 3x^2 \).
4. Step 4: Simplify the expression \( (f-g)(x) = x - 5 - 3x^2 \). Rearrange the terms to get \( (f-g)(x) = -3x^2 + x - 5 \).
5. Step 5: Determine the domain of \( (f-g)(x) \). Since both \( f(x) \) and \( g(x) \) are polynomials, their domain is all real numbers. Therefore, the domain of \( (f-g)(x) \) is also all real numbers.

Key Concepts

Function Subtraction

Function subtraction involves taking two functions, f(x) and g(x), and creating a new function, f-g, defined as (f-g)(x) = f(x) - g(x). In this case, you will subtract the output of g(x) from the output of f(x) for each x in the domain of both functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like f(x) = x - 5 and g(x) = 3x², the domain is typically all real numbers unless specified otherwise by restrictions such as square roots or denominators.

Combining Functions

When combining functions, such as through addition, subtraction, or multiplication, it is essential to consider the domains of the individual functions. The domain of the resulting function (f-g) will be the intersection of the domains of f and g, ensuring that the operation is valid for all x in that domain.