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Chapter 2: Problem 48

Determine if the function is even, odd, or neither. $$ g(x)=-x $$

Short Answer

Expert verified
The function \( g(x) = -x \) is odd.

Step by step solution

01

- Understand the definitions

Recall the definitions of even and odd functions. A function is even if \(f(x) = f(-x)\) for all x in its domain. A function is odd if \(f(-x) = -f(x)\) for all x in its domain.
02

- Evaluate the function at \( x \)

Given the function \( g(x) = -x \), evaluate it at \( x \).
03

- Evaluate the function at \( -x \)

Now, evaluate the function at \( -x \): \( g(-x) = -(-x) = x \).
04

- Compare \( g(x) \) and \ (g(-x) \)

Compare the values found: \( g(x) = -x \) and \( g(-x) = x \). Notice that \( g(-x) = -g(x) \).
05

- Determine the function type

Since \( g(-x) = -g(x) \, the function \ g(x) = -x \) satisfies the condition for an odd function. Thus, the function is odd.

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Key Concepts

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

Functions in Algebra
Functions are fundamental in algebra. A function maps an input to an output according to a specific rule. For example, if you have a function \( f(x) \), it means taking a value \( x \) and processing it through the function to get \( f(x) \). Understanding functions is key for many algebraic problems.

To determine if a function is even, odd, or neither, it鈥檚 important to understand how a function behaves when its input is negated. This involves plugging in \( -x \) for \( x \) and comparing the results.
Even Functions
Even functions have a symmetric property. They look the same on both sides of the y-axis when graphed.

The mathematical way to express this is by saying that a function \( f(x) \) is even if:
  • \( f(x) = f(-x) \) for all \( x \) in its domain.


As an example, the function \( f(x) = x^2 \) is even because plugging in \( -x \) yields the same result as plugging in \( x \):
  • \( f(-x) = (-x)^2 = x^2 \).
Odd Functions
Odd functions have rotational symmetry about the origin. Graphically, they look the same after a 180-degree rotation around the origin.

Mathematically, a function \( f(x) \) is odd if:
  • \( f(-x) = -f(x) \) for all \( x \) in its domain.


For instance, if you have \( f(x) = x^3 \), substituting \( -x \) will result in:
  • \( f(-x) = (-x)^3 = -x^3 \).
  • Since \( f(-x) = -f(x) \), \( f(x) = x^3 \) is an odd function.
In the given exercise, we determined that \( g(x) = -x \) is odd. Evaluating \( g(-x) \) gave us \( x \), which translates to \( -g(x) \), thus satisfying the condition for an odd function.
Algebraic Properties
Algebraic properties help us determine the behavior of functions and verify their type. To summarize:
  • **Symmetry**: Even functions show symmetry about the y-axis, whereas odd functions exhibit rotational symmetry about the origin.
  • **Substitution**: By substituting \( -x \) and comparing with the original function, you can determine if a function is even or odd.
  • **Results Analysis**: An even function will always equate \( f(x) \) to \( f(-x) \), while an odd function equates \( f(-x) \) to \( -f(x) \).
These properties provide a useful framework for analyzing functions and understanding their characteristics. Applying these principles can simplify your work with algebraic functions and enhance your understanding of their behavior.

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Most popular questions from this chapter

Determine if the function is even, odd, or neither. $$ k(x)=13 x^{3}+12 x $$

a. Graph \(a(x)=x\) for \(x<1\). b. Graph \(b(x)=\sqrt{x-1}\) for \(x \geq 1\). c. Graph \(c(x)=\left\\{\begin{array}{ll}x & \text { for } x<1 \\ \sqrt{x-1} & \text { for } x \geq 1\end{array}\right.\)

Use a graphing utility to graph the piecewise-defined function. $$ f(x)=\left\\{\begin{array}{ll} 2.5 x+2 & \text { for } x \leq 1 \\ x^{2}-x-1 & \text { for } x>1 \end{array}\right. $$

A car starts from rest and accelerates to a speed of \(60 \mathrm{mph}\) in \(12 \mathrm{sec} .\) It travels \(60 \mathrm{mph}\) for \(1 \mathrm{~min}\) and then decelerates for 20 sec until it comes to rest. The speed of the car \(s(t)\) (in mph) at a time \(t\) (in sec) after the car begins motion can be modeled by: \(s(t)=\left\\{\begin{array}{cl}\frac{5}{12} t^{2} & \text { for } 0 \leq t \leq 12 \\ 60 & \text { for } 12

Evaluate the function for the given values of \(x\). \(f(x)=\left\\{\begin{aligned}-3 x+7 & \text { for } x<-1 \\ x^{2}+3 & \text { for }-1 \leq x<4 \\ 5 & \text { for } x \geq 4 \end{aligned}\right.\) a. \(f(3)\) b. \(f(-2)\) c. \(f(-1)\) d. \(f(4)\) e. \(f(5)\)
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