/*! 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 64 Determine whether the graph of e... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 64

Determine whether the graph of each function is symmetric about the y-axis or the origin. Indicate whether the function is even, odd, or neither. $$ f(x)=(x-1)^{2} $$

Short Answer

Expert verified
The function is neither even nor odd.

Step by step solution

01

- Define Even Function

A function is even if for every x in the function's domain, \[f(-x) = f(x)\]. This can indicate symmetry about the y-axis.
02

- Define Odd Function

A function is odd if for every x in the function's domain, \[f(-x) = -f(x)\]. This can indicate symmetry about the origin.
03

- Calculate \(f(-x)\)

Given the function \(f(x) = (x-1)^2\), replace x with -x:\[f(-x) = (-x-1)^2\]
04

- Simplify \(f(-x)\)

Simplify the expression \((-x-1)^2\):\[(-x-1)^2 = (x+1)^2\]
05

- Compare \(f(-x)\) and \(f(x)\)

Compare the simplified \(f(-x)\) to \(f(x)\):\[f(x) = (x-1)^2\] and \[f(-x) = (x+1)^2\]. Since \((x+1)^2 eq (x-1)^2\), the function is neither even nor odd.
06

- Conclusion

Since \(f(-x)\) does not equal \(f(x)\) and \(f(-x)\) does not equal \(-f(x)\), the function \(f(x) = (x-1)^2\) is neither even nor odd, and the graph is neither symmetric about the y-axis nor the origin.

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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 mathematically defined by the property that for every x in its domain, \( f(-x) = f(x) \). In other words, flipping x to the negative side doesn't change the value of the function. This results in symmetry about the y-axis.

Imagine folding the graph along the y-axis. If the two sides match perfectly, the function is even.
  • Example: The function \( f(x) = x^2 \) is an even function because \( f(-x) = (-x)^2 = x^2 = f(x) \).
Checking for even functions:
  • Replace x with -x in the function.
  • Simplify and see if you get back the original function.
If yes, the function is even. In our exercise, we saw that \( f(x) = (x-1)^2 \) did not satisfy this criteria because \( f(-x) = (x+1)^2 != (x-1)^2 \), hence it's not even.
Odd Function
An odd function is defined by \( f(-x) = -f(x) \), meaning the function returns the negative value of what it was originally when x is replaced by -x.
Said another way, rotating the function 180 degrees around the origin should yield the same graph.
  • Example: The function \( f(x) = x^3 \) is an odd function because \( f(-x) = (-x)^3 = -x^3 = -f(x) \).
To test if a function is odd:
  • Replace x with -x in the function.
  • Simplify and see if you get \ -f(x) \.
If so, the function is odd. In our exercise, \( f(x) = (x-1)^2 \) did not satisfy this condition as well because \( f(-x) = (x+1)^2 != -(x-1)^2 = -f(x) \). Hence, it's not odd either.
Symmetry in Graphs
Symmetry in graphs helps us understand function properties and their behavior visually.

There are two main types of symmetry usually discussed:
  • Symmetry about the y-axis: If you can fold the graph along the y-axis and the two halves match, the function has y-axis symmetry. This is connected to even functions since \( f(-x) = f(x) \).
  • Symmetry about the origin: If rotating the graph 180 degrees around the origin doesn't change its appearance, the function has origin symmetry. This is connected to odd functions where \( f(-x) = -f(x) \).
Why symmetry matters:
  • It simplifies graphing and understanding function behaviors.
  • Helps in predicting future values and behaviors without extensive calculations.
In the context of our exercise, \( f(x) = (x-1)^2 \) did not demonstrate either type of symmetry since it was neither even nor odd.

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