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Chapter 3: Problem 5

Classfy each function as odd, even, or neither. $$g(x)=-x$$

Short Answer

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

Step by step solution

01

Test for evenness

To find out if the function \(g(x)\) is even, replace \(x\) with \(-x\) in the function's formula and see if the resulting expression is equal to original formula \(g(x)\). So, compute \(g(-x)\) and compare with \(g(x)\). For the given function \(g(x)=-x\), replacing \(x\) with \(-x\), the function becomes \(g(-x)=-(-x) = x\). Clearly, \(g(-x) \neq g(x)\), hence the function is not even.
02

Test for oddness

Now, to confirm if the function \(g(x)\) is odd, compare \(-g(x)\) with \(g(-x)\). As we already know that \(g(-x) = x\), we will now find \(-g(x)\). For the given function \(g(x)=-x\), \(-g(x)=-(-x)=x\). Since \(-g(x)\) equals \(g(-x)\) the function is odd.

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Key Concepts

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

Even Function Test
An even function is one that, when evaluated at both a variable and its negative, yields the same result: \( f(x) = f(-x) \). This symmetry resembles a reflection over the y-axis.
To determine whether a function is even, you replace every instance of \( x \) in the function with \(-x\). Then, check if the resulting expression equals the original function formula:
For example, given a function \( g(x) = -x \), let's perform the even function test:
  • Substitute \(-x\) for \(x\): \( g(-x) = -(-x) = x \)
  • Compare: Since \( g(x) = -x \) and \( g(-x) = x \), clearly \( g(-x) eq g(x) \)
Therefore, \( g(x) = -x \) is not an even function. Identifying even functions helps in understanding their graphs and properties.
Odd Function Test
Odd functions have a distinct characteristic: they satisfy the condition \( f(-x) = -f(x) \). This property reflects the function's symmetry about the origin, which means rotating 180掳 around the origin yields the same graph.
To determine if a function is odd:
  • Calculate \( g(-x) \), as before.
  • Compute the negative of the original function, \(-g(x)\).
  • Compare the two results:
Using the function \( g(x) = -x \), check if \( g(-x) = -g(x) \):
  • We already found \( g(-x) = x \).
  • Calculate \(-g(x) = -(-x) = x \).
  • Since \( g(-x) = x \) and \(-g(x) = x \), the function is confirmed as odd because \( g(-x) = -g(x) \).
Recognizing odd functions helps with predicting their behavior and analyzing their unique symmetric properties.
Function Classification
Classifying functions into even, odd, or neither is a fundamental task. It helps in simplifying problems and understanding graphs.
A quick reminder of definitions:
  • Even functions have symmetry over the y-axis: \( f(x) = f(-x) \).
  • Odd functions rotate symmetrically around the origin: \( f(-x) = -f(x) \).
  • If neither condition holds, the function is neither even nor odd.
With the function \( g(x) = -x \), we determined it as odd because:
  • Substituting \(-x\) led to \( g(-x) = x \), not equal to \( g(x) = -x \), failing the even function test.
  • However, \(-g(x) = x\) was equal to \( g(-x) \), passing the odd function test.
This classification helps in expectation setting when mapping function graphs and during calculus operations that engage these properties.

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Most popular questions from this chapter

Give a possible expression for a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=2\) and a vertical asymptote \(x=1,\) with \(y\) intercept at \((0,0) .\) It may be helpful to sketch the graph of \(r\) first. You may check your answer with a graphing utility.

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