In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 16

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the function $f-g$, which means we will subtract the function $g(x)$ from $f(x)$.
2. Step 2: Write down the expressions for $f(x)$ and $g(x)$. We have $f(x)=3-x^2$ and $g(x)=x^2+2x-16$.
3. Step 3: Subtract $g(x)$ from $f(x)$. This means we calculate $(3-x^2)-(x^2+2x-16)$.
4. Step 4: Distribute the negative sign across the terms in $g(x)$. This gives us $3-x^2-x^2-2x+16$.
5. Step 5: Combine like terms to simplify the expression. This will give you the function $f-g$.

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). This operation requires substituting the expressions for f(x) and g(x) into the equation and simplifying the result to find the new function.

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) and g(x), the domain is typically all real numbers, but when combining functions, it is essential to ensure that the resulting function does not introduce any restrictions.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this case, both f(x) and g(x) are quadratic polynomials, which means they can be graphed as parabolas. Understanding their shapes and behaviors is crucial for analyzing their differences and determining the domain.