Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2 +1 b. h (-1)
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 31a
Find ƒ+g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1
Verified step by step guidance1
Identify the given functions: \(f(x) = 2x + 3\) and \(g(x) = x - 1\).
To find \((f + g)(x)\), add the two functions together: \((f + g)(x) = f(x) + g(x)\).
Substitute the expressions for \(f(x)\) and \(g(x)\) into the sum: \((f + g)(x) = (2x + 3) + (x - 1)\).
Combine like terms to simplify the expression: \((f + g)(x) = 2x + x + 3 - 1\).
Determine the domain of \((f + g)(x)\) by finding the intersection of the domains of \(f(x)\) and \(g(x)\). Since both are linear functions defined for all real numbers, the domain is all real numbers.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Addition (ƒ + g)
Function addition involves creating a new function by adding the outputs of two given functions for the same input value. For functions f(x) and g(x), (ƒ + g)(x) = f(x) + g(x). This operation combines the expressions algebraically to form a single function.
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Function Composition
Domain of a Function
The domain of a function is the set of all input values (x-values) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all inputs are valid for both.
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Linear Functions
Linear functions are polynomial functions of degree one, typically written as f(x) = mx + b. They have a constant rate of change and their graphs are straight lines. Understanding their form helps in performing operations like addition and determining domains easily.
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Related Practice
Textbook Question
In Exercises 31–32, the domain of each piecewise function is (-∞, ∞) (a) Graph each function. (b) Use the graph to determine the function's range.
Textbook Question
Write the standard form of the equation of the circle with the given center and radius. Center (0, 0), r = 7
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Textbook Question
Which graphs in Exercises 29–34 represent functions that have inverse functions?
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Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2+1 a. h (2)
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Textbook Question
Find f−g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1