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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 1 - Functions문제 1.1.40
1장, 문제 1.1.40

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = 1/|x|

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First, understand the function y = 1/|x|. The absolute value function |x| affects the graph by making all x-values positive, which means the function is defined for all x except x = 0.
Consider the symmetry of the function. Since |x| is symmetric about the y-axis, the function y = 1/|x| is also symmetric about the y-axis. This means the graph will look the same on both sides of the y-axis.
Next, analyze the behavior of the function as x approaches 0 from the left and right. As x approaches 0, |x| becomes very small, making 1/|x| very large. Therefore, the function has a vertical asymptote at x = 0.
Determine the intervals of increase and decrease. For x > 0, as x increases, |x| increases, making 1/|x| decrease. Thus, the function is decreasing on the interval (0, ∞). For x < 0, as x decreases (approaches 0 from the left), |x| decreases, making 1/|x| increase. Thus, the function is increasing on the interval (-∞, 0).
Finally, graph the function. Plot points for various values of x to see the behavior of the function. Remember the symmetry about the y-axis and the vertical asymptote at x = 0. The graph will show the function decreasing on (0, ∞) and increasing on (-∞, 0), with symmetry about the y-axis.

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주요 개념

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Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing on an interval if the function's value at the second point is less than at the first. Identifying these intervals is crucial for understanding the behavior of the function and can be determined by analyzing the first derivative.
추천 영상:
07:32
Determining Where a Function is Increasing & Decreasing

Graphing Functions

Graphing a function involves plotting its values on a coordinate plane, which visually represents its behavior. For the function y = 1/|x|, the graph will show how the function behaves as x approaches zero and as x moves away from zero. Understanding the graph helps in identifying symmetries and the intervals of increase and decrease.
추천 영상:
5:53
Graph of Sine and Cosine Function

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits a mirror-like behavior about a specific axis. For example, a function is even if f(x) = f(-x) for all x, indicating symmetry about the y-axis. In the case of y = 1/|x|, the graph is symmetric about the y-axis, which can simplify the analysis of its increasing and decreasing intervals.
추천 영상:
06:21
Properties of Functions
관련 실천
교과서 질문

In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = |x| - 2

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Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


h(t) = |t³|

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교과서 질문

Evaluate cos (11π/12) as cos (π/4 + 2π/3).

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Theory and Examples


The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

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교과서 질문

Using the Half-Angle Formulas


Find the function values in Exercises 47–50.


cos² 5π/12

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교과서 질문

In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = 1 - cos x

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