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Chapter 1: Problem 10

Sketch a graph of an odd function and give the function's defining property.

Short Answer

Expert verified
Answer: The defining property of an odd function is that f(-x) = -f(x) for all x in its domain, which indicates point symmetry with respect to the origin. The graph of f(x) = x^3 demonstrates this property by having a 180-degree rotational symmetry around the origin, resembling a stretched letter "S" with the left side descending downwards while the right side ascends upwards.

Step by step solution

01

Understand the defining property of odd functions

An odd function is a function that satisfies the property f(-x) = -f(x) for all x in its domain. In simpler words, an odd function has a point symmetry with respect to the origin: if you rotate the graph of the function 180 degrees around the origin, it will remain the same.
02

Choose a simple odd function

There are many odd functions to choose from. For this exercise, we'll use a simple one that can be easily graphed: f(x) = x^3. This function has the property that f(-x) = -f(x) for all x because f(-x) = (-x)^3 = -x^3 = -f(x).
03

Sketch the graph

To sketch the graph of f(x) = x^3, we'll plot a few points and observe the behavior of the curve. Some points to include are f(-2)=-8, f(-1)=-1, f(0)=0, f(1)=1, and f(2)=8. The graph of the function starts in the third quadrant at a very low negative value and moves up and towards the second quadrant as x approaches 0. At the origin, it passes through (0,0) and then continues to rise into the first quadrant as x increases, reaching very high positive values when x is large. To emphasize the odd function properties, you can mark the origin, and add dashed lines to show the 180-degree rotational symmetry. The overall shape of the graph is similar to a stretched letter "S", with the left side descending downwards and the right side ascending upwards, maintaining symmetry around the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Symmetry
Understanding function symmetry is crucial when analyzing graphs. Symmetry in functions refers to a situation where parts of the graph reflect or rotate in predictable ways:
  • Even Symmetry: A function has even symmetry if it mirrors across the y-axis. Mathematically, this means that the function satisfies the property \( f(x) = f(-x) \).
  • Odd Symmetry: Odd symmetry, the focus of this exercise, refers to a point symmetry around the origin, ensuring \( f(-x) = -f(x) \). This means that if you rotate the graph 180 degrees around the origin, it appears unchanged.
Recognizing these symmetrical properties helps in sketching graphs quickly and identifying function behavior.
Graph Sketching
Sketching graphs of functions often involves recognizing patterns and key points, allowing visualization of the function's behavior.
To graph an odd function like \( f(x) = x^3 \):
  • Start by identifying critical points on the function, such as where \( f(x) = 0 \). For \( f(x) = x^3 \), this point is at the origin (0,0).
  • Choose additional points such as \( x = 1 \), \( -1 \), \( 2 \), and \( -2 \) to understand its curvature on different parts of the function.
  • Note the pattern: as \( x \) moves from negative to positive, the function transitions smoothly from negative to positive, through the origin.
  • The graph resembles an "S" shape; its symmetry about the origin is visually apparent.
This approach provides a structured way to sketch any odd function.
Point Symmetry
Point symmetry is a type of symmetry where one part of a figure or graph matches another, after a rotation about a central point.For odd functions, the central point is the origin (0,0), and this symmetry results in the graph looking the same after a 180-degree rotation.
For the function \( f(x) = x^3 \):
  • Start with point (1,1). After a 180-degree rotation about the origin, this point moves to (-1,-1).
  • Point (-2,-8) rotates to (2,8).
  • Each point \((x, f(x))\) corresponds with \((-x, -f(x))\), creating the needed symmetry.
  • The graph passes through the origin, a characteristic of many odd functions.
Recognizing point symmetry is essential for verifying whether a function is odd.
Defining Property of Functions
The defining property of mathematical functions reveals key characteristics.For odd functions, this is expressed as \( f(-x) = -f(x) \).This property signifies that:
  • Each value \( x \) and its negative \( -x \) have outputs that are also negatives of each other.
  • The degree of a polynomial function often indicates whether it's odd. For example, cubic functions, such as \( x^3 \), typically meet the odd function criteria.
  • This property assures that odd functions demonstrate symmetry about the origin, vital for analyzing behavior and graph shape.
Understanding the defining properties helps identify relationships in function behavior, aiding both calculation and graph interpretation.

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Most popular questions from this chapter

Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is \(600 \mathrm{m}\) from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x .\) Graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x\). Graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?

(Torricelli's law) A cylindrical tank with a cross-sectional area of \(100 \mathrm{cm}^{2}\) is filled to a depth of \(100 \mathrm{cm}\) with water. At \(t=0,\) a drain in the bottom of the tank with an area of \(10 \mathrm{cm}^{2}\) is opened, allowing water to flow out of the tank. The depth of water in the tank at time \(t \geq 0\) is \(d(t)=(10-2.2 t)^{2}\) a. Check that \(d(0)=100,\) as specified. b. At what time is the tank empty? c. What is an appropriate domain for \(d ?\)

Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|}\hline x & y \\\\\hline 0 & -1 \\\\\hline 1 & 0 \\\\\hline 4 & 1 \\\\\hline 9 & 2 \\\\\hline 16 & 3 \\ \hline\end{array}$$

Let \(E\) be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. $$\boldsymbol{O} \circ \boldsymbol{E}$$

Assume that \(b > 0\) and \(b \neq 1\). Show that \(\log _{1 / b} x=-\log _{b} x\).
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