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Chapter 0: Problem 20

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$g(u)=\frac{u^{3}}{8}$$

Short Answer

Expert verified
The function is odd. Its graph is symmetric around the origin.

Step by step solution

01

Understand the Definition of Even and Odd Functions

An **even function** satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). Graphically, even functions are symmetric around the y-axis. An **odd function** satisfies \( f(-x) = -f(x) \). Its graph is symmetric around the origin.
02

Apply the Even-Odd Test

For \( g(u) = \frac{u^3}{8} \), calculate \( g(-u) \) to test for symmetry. Compute:\[g(-u) = \frac{(-u)^3}{8} = \frac{-u^3}{8} = -\left(\frac{u^3}{8}\right)\]Since \( g(-u) = -g(u) \), the function \( g \) is odd.
03

Sketch the Graph of g(u)

The graph of \( g(u) = \frac{u^3}{8} \) is a cubic function scaled vertically by \(\frac{1}{8}\). Because it is odd, it is symmetric with respect to the origin. The basic shape is an 'S' curve passing through the origin (0,0), sliding smoothly upwards for positive \( u \) and downwards for negative \( u \). Points like \( (1, \frac{1}{8}) \) and \( (-1, -\frac{1}{8}) \) help in plotting.

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Key Concepts

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

even function
Even functions are a special type of function characterized by symmetry. Specifically, an even function satisfies the equivalence condition: \( f(-x) = f(x) \) for every value of \( x \) in its domain. This means that the left and right halves of the function mirror each other across the y-axis.

  • Even functions will always create a symmetric shape around the y-axis.
  • This symmetry implies that every point on the curve has a corresponding point directly opposite to it on the other side of the y-axis.
  • An example of a well-known even function is \( f(x) = x^2 \), which forms a parabola opening upwards.


Understanding the nature of even functions can help in predicting the graph's appearance and confirming the function type.

Whenever you face an unknown function, checking this condition can quickly tell you if it's even, giving you a useful property of its graph.
function symmetry
Function symmetry is a concept used to describe how a function behaves when transformed or flipped across a line or a point.

Symmetry in functions usually appears in two primary forms: y-axis symmetry and origin symmetry.

Sometimes, functions can have other types of symmetry depending on transformations applied, but these are less common.

Understanding symmetry in functions greatly aids in sketching and analyzing their graphs.

**Y-Axis Symmetry**
  • A function is symmetric with respect to the y-axis if it is even, satisfying \( f(-x) = f(x) \).
  • This type of symmetry implies that the graph on the positive side of x reflects the graph on the negative side.

**Origin Symmetry**
  • Origin symmetry is a characteristic of odd functions, where \( f(-x) = -f(x) \).
  • This means the function's graph looks the same when rotated 180 degrees around the origin.
  • In essence, if a function passes through the origin and behaves the same in all quadrants, the function is odd.
graph of a function
The graph of a function offers a visual representation of all the outputs the function can achieve from its inputs.

Creating and analyzing these graphs is essential in understanding the function's behavior.

**Sketching a Graph**
When sketching a graph, you begin by understanding the function's formula and essential features. Here are some steps to guide you:

  • Identify key properties such as domain, range, and intercepts. For example, find points where the function crosses the x-axis or y-axis.
  • Evaluate critical points and derivatives if applicable to determine slopes and behavior changes.
  • Check for symmetry, as it simplifies the sketching process.
  • Consider behavior at the limits, seeing how the function acts as \( x \) approaches infinity or negative infinity.

**Understanding Graph Shapes**
Different functions yield various shapes based on their exponents and roots:

  • Linear functions yield straight lines, described by \( f(x) = mx + b \).
  • Quadratic functions produce parabolas, typically in the form \( f(x) = ax^2 + bx + c \).
  • Cubic functions, like \( g(u) = \frac{u^3}{8} \), create S-shaped curves due to their characteristic polynomial degree.

By understanding these principles, you will become more adept at determining how a function behaves through its graphical form.

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Most popular questions from this chapter

Among all lines perpendicular to \(4 x-y=2,\) find the equation of the one that, together with the positive \(x\) - and \(y\) -axes, forms a triangle of area 8 .

A regular polygon of \(n\) sides is inscribed in a circle of radius \(r\). Find formulas for the perimeter, \(P,\) and area, \(A,\) of the polygon in terms of \(n\) and \(r\).

Find the area of the sector of a circle of radius 5 centimeters and central angle 2 radians (see Problem 42 ).

Graph the function \(f(x)=\sin 50 x\) using the window given by a \(y\) range of \(-1.5 \leq y \leq 1.5\) and the \(x\) range given by (a) [-15,15] (b) [-10,10] (c) [-8,8] (d) [-1,1] (e) [-0.25,0.25] Indicate briefly which \(x\) -window shows the true behavior of the function, and discuss reasons why the other \(x\) -windows give results that look different.

Circular motion can be modeled by using the parametric representations of the form \(x(t)=\sin t\) and \(y(t)=\cos t\) (A parametric representation means that a variable, \(t\) in this case, determines both \(x(t)\) and \(y(t) .\) This will give the full circle for \(0 \leq t \leq 2 \pi .\) If we consider a 4 -foot-diameter wheel making one complete rotation clockwise once every 10 seconds, show that the motion of a point on the rim of the wheel can be represented by \(x(t)=2 \sin (\pi t / 5)\) and \(y(t)=2 \cos (\pi t / 5)\) (a) Find the positions of the point on the rim of the wheel when \(t=2\) seconds, 6 seconds, and 10 seconds. Where was this point when the wheel started to rotate at \(t=0 ?\) (b) How will the formulas giving the motion of the point change if the wheel is rotating counterclockwise. (c) At what value of \(t\) is the point at (2,0) for the first time?
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