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Chapter 2: Problem 50

Based on the ordered pairs seen in each table, make a conjecture about whether the function \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r} x & f(x) \\ \hline-3 & -5 \\ -2 & -4 \\ -1 & -1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 5 \end{array}$$

Short Answer

Expert verified
The function is odd.

Step by step solution

01

Understand Even and Odd Functions

Recall that a function is even if for every \(x\), \(f(x) = f(-x)\), and it is odd if for every \(x\), \(f(x) = -f(-x)\). We will use these definitions to check the conditions of even and odd functions for the given data.
02

Evaluate for Even Function

Check whether each pair of \(x\) and \(-x\) satisfies the condition \(f(x) = f(-x)\). From the table, corresponding pairs are \((-3,3)\), \((-2,2)\), \((-1,1)\), and \((0,0)\). Calculate their function values: \(f(-3) = -5\), \(f(3) = 5\); \(f(-2) = -4\), \(f(2) = 4\); \(f(-1) = -1\), \(f(1) = 1\); \(f(0) = 0\). Since none of these pairs satisfy \(f(x) = f(-x)\), the function is not even.
03

Evaluate for Odd Function

Check whether each pair of \(x\) and \(-x\) satisfies the condition \(f(x) = -f(-x)\). From the given pairs: \(-f(-3) = 5\) for \(f(3) = 5\); \(-f(-2) = 4\) for \(f(2) = 4\); \(-f(-1) = 1\) for \(f(1) = 1\). These calculations show that \(f(x) = -f(-x)\) holds for every evaluated pair, except \(f(0)=0\) which is not affected by this condition since zero aligns with both even and odd conditions without contradiction.
04

Make a Conclusion

Since the function satisfies \(f(x) = -f(-x)\) for all \(x\), the function is odd. Therefore, by definition, the function \(f\) is odd.

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Key Concepts

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

Even Function
An even function is a type of mathematical function where the output is symmetric with respect to the y-axis. To determine if a function is even, we check if substituting \(x\) into the function gives the same result as substituting \(-x\), thus the condition \(f(x) = f(-x)\) must hold for all values of \(x\).

**Characteristics of Even Functions**:
  • The graph of an even function reflects across the y-axis.
  • If you know the function value at positive \(x\), you know the value at negative \(x\) because they are the same.
  • Examples of even functions include simple ones like \(f(x) = x^2\) or \(f(x) = \, \cos(x)\).
In the exercise, by checking the ordered pairs, we established that this symmetry does not apply, hence the function in question is not even.
Function Evaluation
Function evaluation involves substituting a given value into a function to determine the output. It's the process of computing the value of the function for a specific input.

**How to Evaluate Functions**:
  • Identify the input value (\(x\)).
  • Substitute this value into the function (\(f\)).
  • Solve the equation to find the output (\(f(x)\)).
For example, in the original exercise, to evaluate the function at \(x = 2\), we look at the table and find that \(f(2) = 4\). This process helps us to analyze and compare values of \(f(x)\) and \(f(-x)\) to ascertain if the function might be even or odd.
Ordered Pairs
Ordered pairs are pairs of numbers used to represent points on a coordinate plane, typically written as \( (x, y) \). When dealing with functions, these pairs allow us to easily see the relationship between inputs (\\(x\)) and their corresponding outputs (\\(f(x)\)).

**Understanding Ordered Pairs**:
  • In a function, each ordered pair links an input to exactly one output.
  • The first number represents the input (or independent variable), and the second number indicates the corresponding output (dependent variable).
  • Analyzing tables of ordered pairs helps in identifying whether a function is even, odd, or neither by applying definitions of even and odd functions.
When the exercise requires checking if a function is even or odd, we use the ordered pairs \( (x, f(x)) \) and \(( -x, f(-x))\) to verify functional properties.
Conjecture in Mathematics
In mathematics, a conjecture is an educated guess or hypothesis based on observation, which is yet to be proven true or false. Conjectures provide a starting point for testing and exploration.

**Making a Conjecture**:
  • Observe patterns and relationships in data.
  • Form a tentative statement or prediction about these observations.
  • Test the conjecture with further examples or through a formal proof.
In the context of the exercise, by observing the pattern in the given table of values, a conjecture was made about the function being odd, which was subsequently confirmed through checking the condition \(f(x) = -f(-x)\) for the given pairs. If all provided ordered pairs meet the odd function condition, which they do except zero, this supports the conjecture that the function is indeed odd.

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Most popular questions from this chapter

For each function find \(f(x+h)\) and \(f(x)+f(h)\). $$f(x)=x^{3}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=5, \quad g(x)=x$$

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An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|0.40 x+2|=|0.60 x-5|$$
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