In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the sum of two functions, $f(x)=3-x^2$ and $g(x)=x^2+2x-15$.
2. Step 2: Add the functions together. The sum $(f+g)(x)$ is found by adding $f(x)$ and $g(x)$ together: $(f+g)(x)=(3-x^2)+(x^2+2x-15)$.
3. Step 3: Simplify the expression. Combine like terms in the expression: $(3-x^2)+(x^2+2x-15)=3-x^2+x^2+2x-15$.
4. Step 4: Further simplify the expression. The $-x^2$ and $x^2$ terms cancel each other out, leaving $3+2x-15$.
5. Step 5: Simplify the remaining terms. Combine the constant terms: $3-15=-12$, so the final expression is $2x-12$. Determine the domain, which is all real numbers since there are no restrictions on $x$ in the expression $2x-12$.

Key Concepts

Function Addition

Function addition involves combining two functions by adding their outputs for each input. For functions f(x) and g(x), the sum is defined as (f + g)(x) = f(x) + g(x). This operation requires evaluating both functions at the same x-value and summing the results, which is essential for solving the given problem.

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) = 3 - x² and g(x) = x² + 2x - 15, the domain is typically all real numbers, as polynomials do not have restrictions such as division by zero or square roots of negative numbers.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The functions f(x) and g(x) in the problem are both polynomials, which means they can be easily added and analyzed for their properties, such as their domain and behavior across the real number line.