Textbook Question
Find fg and determine the domain for each function. f(x) = x -5, g(x) = 3x²
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3. Functions
Function Operations
1:43 minutesProblem 37a
Textbook Question
Find ƒ+g, ƒ- g, ƒg and ƒ/g. Determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15
Verified step by step guidance1
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 (-∞, ∞).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey 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.
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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.
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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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