Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

문제 8

4장, 문제 8

Provide a short answer to each question. Is ƒ(x)=1/x an even or an odd function? What symmetry does its graph exhibit?

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Recall the definitions: A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in the domain, and it is odd if $ f(-x) = -f(x) $ for all $ x $ in the domain.

Start by finding $ f(-x) $ for the given function $ f(x) = \frac{1}{x} $. Substitute $ -x $ into the function: $ f(-x) = \frac{1}{-x} = -\frac{1}{x} $.

Compare $ f(-x) $ with $ f(x) $ and $ -f(x) $: since $ f(-x) = -\frac{1}{x} = -f(x) $, the function satisfies the condition for being an odd function.

Because $ f(x) $ is odd, its graph exhibits symmetry about the origin. This means if you rotate the graph 180 degrees around the origin, it looks the same.

Summarize: $ f(x) = \frac{1}{x} $ is an odd function and its graph has origin symmetry.

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Even and Odd Functions

A function is even if f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. A function is odd if f(-x) = -f(x), indicating symmetry about the origin. Determining whether a function is even or odd helps understand its symmetry properties.

Function Symmetry

Symmetry in functions refers to how their graphs mirror across certain lines or points. Even functions have y-axis symmetry, while odd functions have origin symmetry. Recognizing symmetry helps in graphing and analyzing function behavior efficiently.

Reciprocal Function Characteristics

The function f(x) = 1/x is a reciprocal function defined for all x ≠ 0. It is known to be an odd function because substituting -x yields f(-x) = -1/x = -f(x). Its graph exhibits origin symmetry, with branches in the first and third quadrants.

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Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months. Sketch a graph of $y = V (x)$ for January through December. In what month are the fewest volunteers available?

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교과서 질문

Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

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교과서 질문

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=2x⁴

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