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Chapter 0: Problem 73

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ g(x)=x^{3}-5 x $$

Short Answer

Expert verified
The function \(g(x) = x^{3} - 5x\) is odd and symmetric about the origin.

Step by step solution

01

Setting up the Function

First, define the function \(g(x) = x^{3} - 5x\).
02

Testing for Evenness

Calculate the value of \(g(-x)\) and compare it with \(g(x)\). If \(g(-x) = g(x)\) for all x in the function's domain, then the function is even. \[ g(-x) = (-x)^{3} - 5(-x) = -x^{3} + 5x. \] As \(g(-x) \neq g(x)\), the function is not even.
03

Testing for Oddness

Now, compare \(g(-x)\) with \(-g(x)\) to test for oddness. If \(g(-x) = -g(x)\) for all x in the function's domain, then the function is odd. \[ -g(x) = -(x^{3} - 5x) = -x^{3} + 5x. \] Since \(g(-x) = -g(x)\), the function is odd.
04

Describing the Symmetry

Odd functions are symmetric about the origin, so the function \(g(x) = x^{3} - 5x\) is symmetric about the origin.

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Key Concepts

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

Function Symmetry
In mathematics, understanding function symmetry can simplify graph sketches and provide insights into the properties of functions.

Function symmetry refers to the way a function behaves when you replace its input with its additive inverse. If a function maintains certain characteristics after this substitution, it exhibits symmetry. There are two primary types of symmetry you might encounter: even symmetry and odd symmetry.
  • Even symmetry: A function is considered even if the function satisfies the condition: \[ f(-x) = f(x) \] for every x in the domain of the function. Graphically, this symmetry means the graph is mirror-image symmetric with respect to the y-axis.
  • Odd symmetry: A function is odd when: \[ f(-x) = -f(x) \] for every x. This means the graph is symmetric with respect to the origin, such as the function \( g(x) = x^3 - 5x \) from our example.
Recognizing these symmetries can be particularly useful when you sketch functions, analyze their properties, or solve equations.
Even and Odd Functions
Functions are classified as even or odd based on specific algebraic properties. Knowing whether a function is even, odd, or neither can help in understanding its behavior and the nature of its symmetry.

In our example, the function \[ g(x) = x^3 - 5x \] can be identified by performing the following operations:
  • Even Function: Evaluate \[ g(-x) \] and compare it to \[ g(x) \]. For original function, we found that: \[ g(-x) = -x^3 + 5x \] which is not equal to \[ g(x) = x^3 - 5x \]. Therefore, this function is not even.
  • Odd Function: Compare \[ g(-x) \] to \[ -g(x) \]. The computation shows \[ -g(x) = -x^3 + 5x \] which matches \[ g(-x) \]. As a result, \[ g(x) \] is an odd function.
Once identified, these classifications make it easier to infer the graph's symmetry, which can assist with further analysis or in graph-based problems.
Graph Symmetry
Graph symmetry, in terms of functions, is an important concept that makes the relationship between algebraic expressions and geometric representations clearer.

When you graph functions, observing symmetry can provide valuable clues:
  • Y-axis Symmetry (Even Functions): Graphs of even functions look the same on both sides of the y-axis because for every point \((x, y)\), the point \((-x, y)\) is also on the graph.
  • Origin Symmetry (Odd Functions): For odd functions, the graph exhibits symmetry about the origin. If a point \((x, y)\) is on the graph, then the point \((-x, -y)\) is too. This explains why our example function \(g(x) = x^3 - 5x\) exhibits origin symmetry, conforming to the properties of odd functions.
Understanding and recognizing graph symmetry can significantly ease the sketching of function graphs and verify solutions to function-related problems.

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Most popular questions from this chapter

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-\sqrt{x+1}+3 &\quad x_{1}=3, x_{2}=8 \end{array} $$

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In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=-\frac{2}{x} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ \(f(x)=-\sqrt{x-2}+5 &\quad x_{1}=3, x_{2}=11\) \end{array} $$
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