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Chapter 1: Problem 68

\(65-70\) Determine whether \(f\) is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. $$f(x)=x|x|$$

Short Answer

Expert verified
The function is odd.

Step by step solution

01

Understand the definition of even and odd functions

A function is even if for every x in the domain, \(f(-x) = f(x)\). A function is odd if for every x in the domain, \(f(-x) = -f(x)\). If neither condition holds, the function is neither even nor odd.
02

Find the negative of the input

Replace \(x\) with \(-x\) in the function to consider the negative input:\[f(-x) = -x|-x|.\] Compute this expression to further determine its nature.
03

Simplify the expression for negative input

Since \(-x\) will be negative when \(x\) is positive, the absolute value \(|-x| = x\), therefore: \[f(-x) = -x|x| = -(x|x|).\] Thus, \(f(-x) = -f(x)\).
04

Determine the function's parity

Since \(f(-x) = -f(x)\), the function \(f(x) = x|x|\) is odd.
05

Confirm the solution using a graphing tool

Use a graphing calculator to plot \(f(x) = x|x|\). The graph should be symmetric about the origin, confirming that 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
An even function is one where the graph has symmetry about the y-axis. This symmetry means that for every point \(x, f(x)\) on the graph, there is a corresponding point \(-x, f(x)\). So, visually, if you were to fold the graph along the y-axis, both sides would match perfectly.
To mathematically confirm if a function is even, you check the formula \(f(-x) = f(x)\). If this equality holds for all x in the domain, then the function is even. Common examples include \(f(x) = x^2\) and \(f(x) = \cos(x)\). These functions exhibit y-axis symmetry and satisfy the condition of being even.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value is denoted as \(|x|\). For instance:
  • \(|5| = 5\)
  • \(|-3| = 3\)
Absolute value plays a key role in understanding functions like \(f(x) = x|x|\), which combines linear terms with absolute value. It captures how a function behaves differently across positive and negative inputs by enforcing non-negativity, thus making an impact on the algebraic manipulation of expressions involving negatives.
Function Symmetry
Function symmetry is a powerful concept in understanding how functions behave. Two major types of symmetry exist:
  • **Y-axis symmetry** suggests that a function is even. This means the graph looks the same on both sides of the y-axis.
  • **Origin symmetry** is characteristic of odd functions. This means that rotating the graph by 180掳 about the origin results in the same graph.
For a function like \(f(x) = x|x|\), showing origin symmetry means confirming \(f(-x) = -f(x)\). This symmetry allows one to predict function behavior without complete graph plotting.
Graphing Calculator
A graphing calculator is a tool that can visually confirm the nature of a function. By plotting \(f(x) = x|x|\) on a graphing calculator, you can check for symmetry. For odd functions, like in the current analysis, the graph should show symmetry about the origin.
Using a graphing calculator to visualize functions gives quick insights into their nature. It supports analytical results by providing a graphical representation, making it easier to see complex behaviors. This visual approach aids in confirming mathematical properties determined through algebra, such as whether a function is even, odd, or neither.

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Most popular questions from this chapter

\(51-52\) Solve each inequality for \(x\) (a) \(2<\ln x<9 \quad\) (b) \(e^{2-3 x}>4\)

\(59-64\) Find the exact value of each expression. (a) \(\cot ^{-1}(-\sqrt{3}) \quad\) (b) arccos \(\left(-\frac{1}{2}\right)\)

\(45-50\) Find an expression for the function whose graph is the given curve. $$\begin{array}{l}{\text {The line segment joining the points } (-5,10) and (7,-10)}\end{array}$$

Graph the function \(y = x ^ { n } 2 ^ { - x } , x \geq 0 ,\) for \(n = 1,2,3,4,5\) and \(6 .\) How does the graph change as n increases?

\(47-49 \text { Express the function in the form } \mathrm{f} \circ g \circ \mathrm{h.}\) \(H(x)=\sqrt[8]{2+|x|}\)
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