/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Determine whether \(f\) is even,... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 62

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 \( f(x) = x|x| \) is odd.

Step by step solution

01

Function Definition

The function given is \( f(x) = x|x| \). Here, \(|x|\) denotes the absolute value of \(x\). The absolute value function is defined as \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). We need to evaluate whether this function is even, odd, or neither.
02

Even Function Test

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \(x\). Let's calculate \( f(-x) \). Substitute \(-x\) in place of \(x\):\[ f(-x) = -x|-x| \]Using the property of absolute values, \(|-x| = |x|\), we get:\[ f(-x) = -x|x| \]For \( f(x) \) to be even, \( f(-x) \) must equal \( x|x| \), but clearly \( -x|x| eq x|x| \) if \( x eq 0\). Therefore, \( f(x) \) is not even.
03

Odd Function Test

A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \(x\). We previously found that:\[ f(-x) = -x|x| \]Now, calculate \(-f(x)\):\[ -f(x) = -(x|x|) = -x|x| \]Since \( f(-x) = -f(x) \), \( f(x) = x|x| \) satisfies the condition for odd functions.
04

Conclusion

Since the condition \( f(-x) = -f(x) \) holds for all \(x\), the function \( f(x) = x|x| \) is an odd function.
05

Visual Verification (Optional)

If you have a graphing calculator, you can graph \( f(x) = x|x| \) to visually check its symmetry. An odd function is symmetric about the origin, meaning it will look the same if rotated 180 degrees around the origin.

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Key Concepts

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

Even and Odd Functions
Understanding whether a function is even, odd, or neither is a fundamental concept in algebra. It helps to predict the behavior and symmetry of the function's graph. Here鈥檚 how you can determine if a function is even or odd:

  • A function is even if for every input value, the output is the same when the input is replaced by its negative. Mathematically, this means: \( f(-x) = f(x) \) for all \( x \).
  • A function is odd if for every input value, the output is the negative of what it is when the input is replaced by its negative. This condition is expressed as: \( f(-x) = -f(x) \) for all \( x \).
  • If neither of these conditions is met, the function is neither even nor odd.
For the exercise at hand, the function \( f(x) = x|x| \) is shown to be odd because the condition \( f(-x) = -f(x) \) is satisfied.
Absolute Value Function
The absolute value function, denoted by \( |x| \), is a basic yet crucial function in mathematics. It represents the "magnitude" or "distance" of a number from zero on the number line and is always non-negative. Here's how the absolute value function works:

  • If \( x \geq 0 \), then \( |x| = x \).
  • If \( x < 0 \), then \( |x| = -x \).
In the given function \( f(x) = x|x| \), the absolute value plays a key role because it affects how the function behaves under sign changes or reflection across the y-axis (or origin, in the case of odd functions). When solving, it's important to evaluate both cases of the absolute value function.
Graphing Calculator
Graphing calculators are excellent tools for visualizing mathematical functions and concepts, including symmetry and transformations. If you can use a graphing calculator, here's how it can help with this type of problem:

  • Input the function \( f(x) = x|x| \) into the calculator.
  • Use the graphing feature to sketch the function's graph.
  • Observe the graph for aspects of symmetry. An odd function will show symmetry through rotation around the origin, depicted as a consistent curve opposite from each side.
  • This visual feedback provides intuitive confirmation of algebraic findings.
Being able to "see" math in action can greatly enhance understanding and retention.
Function Verification
Verifying the properties of a function involves several systematic steps. It ensures that abstract mathematical definitions are consistent with computational results. Here鈥檚 a brief walkthrough of the function verification process, specifically for even and odd functions:

  • Calculate \( f(-x) \) and compare it algebraically to \( f(x) \) and \(-f(x)\).
  • Determine if \( f(-x) = f(x) \) (for evenness) or \( f(-x) = -f(x) \) (for oddness).
  • For our function \( f(x) = x|x| \), we calculated \( f(-x) = -x|x| \), which matched \(-f(x)\), confirming odd symmetry.
  • As a supplementary check, use visual methods, like a graphing calculator, to corroborate findings.
It's similar to following a recipe. Stick to the steps methodically, and you'll verify the function's characteristics accurately.

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