Find ƒ+g, ƒ- g, ƒg and ƒ/g. Determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

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Step 1: Understand the problem. We are tasked with finding the sum of two functions, ƒ(x) and g(x), denoted as (ƒ + g)(x). This means we need to add the two given functions together and simplify the resulting expression.

Step 2: Write the expressions for ƒ(x) and g(x). The given functions are ƒ(x) = 3 − x² and g(x) = x² + 2x − 15.

Step 3: Add the two functions. Combine ƒ(x) and g(x) by adding their expressions: (ƒ + g)(x) = ƒ(x) + g(x) = (3 − x²) + (x² + 2x − 15).

Step 4: Simplify the resulting expression. Combine like terms: (ƒ + g)(x) = 3 − x² + x² + 2x − 15. The x² terms cancel out, leaving (ƒ + g)(x) = 2x − 12.

Step 5: Determine the domain of the resulting function. Since the resulting function (ƒ + g)(x) = 2x − 12 is a polynomial, it is defined for all real numbers. Therefore, the domain is all real numbers, which can be expressed as (-∞, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 are continuous and smooth, making their behavior predictable across their domains. Understanding their structure is crucial for performing operations like addition.

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