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Chapter 2: Problem 67

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=|x-2|$$

Short Answer

Expert verified
The function \(f(x) = |x-2|\) is neither even nor odd.

Step by step solution

01

Understand the Function

The given function is \(f(x)=|x-2|\). This is an absolute value function, which measures the distance of \(x-2\) from 0.
02

Define Even and Odd Functions

A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\). A function is odd if \(f(-x) = -f(x)\) for all \(x\). If neither condition holds, the function is neither even nor odd.
03

Calculate \(f(-x)\)

Substitute \(-x\) for \(x\) in the function: \[f(-x) = |-x-2|\]
04

Simplify \(f(-x)\)

Simplify the expression: \[f(-x) = |-(x+2)|\]. Since the absolute value of a number negates the negative sign, \(|-(x+2)| = |x+2|\). Thus, \[f(-x) = |x+2|\]
05

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

We have \(f(x) = |x-2|\) and \(f(-x) = |x+2|\). Note that \(|x-2| eq |x+2|\) except for specific values of \(x\). Therefore, \(f(-x) eq f(x)\).
06

Check if Function is Odd

To determine if the function is odd, check if \(f(-x) = -f(x)\). Therefore, we need \[|x+2| = -|x-2|\]. Since the right-hand side results in a negative absolute value, which is not possible, the function cannot be odd. Hence, \(f(-x) eq -f(x)\).
07

Conclusion

\(f(x)\) is neither even nor odd since it fails the conditions for both even and odd functions. As for symmetry, the function does not possess y-axis symmetry (even) or origin symmetry (odd).

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Key Concepts

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

even functions
Even functions have a specific symmetric property. If a function is even, it means that the graph of the function is symmetric with respect to the y-axis. Mathematically, a function \(f(x)\) is considered even if and only if \(f(-x) = f(x)\) for all \(x\). To determine this for any function, replace \(x\) with \(-x)\) and simplify.
Here are a few examples of even functions:
  • The function \(f(x) = x^2\) is even because \(f(-x) = (-x)^2 = x^2\).
  • The cosine function \(f(x) = \cos(x)\) is also even because \(\cos(-x) = \cos(x)\).
If a function is found to satisfy this condition, its graph will mirror itself relative to the y-axis.
odd functions
Odd functions have a different symmetry. A function is odd if the graph is symmetric with respect to the origin. Mathematically, a function \(f(x)\) is considered odd if and only if \(-f(x) = f(-x)\).
Here are a few examples of odd functions:
  • The function \(f(x) = x^3\) is odd because \(-f(x) = -x^3\) and \(f(-x) = (-x)^3\), so \(-x^3 = x^3\).
  • The sine function \(\f(x) = \sin(x)\) is also odd because \(\sin(-x) = -\sin(x)\).
  • Another example is \(f(x) = x\), which is linearly proportional since \(f(-x) = -x\).
Similar to even functions, when a function satisfies the odd property, its graph is symmetric around the origin.
absolute value function
The absolute value function is pivotal when considering symmetry since it removes any negative signs. The basic form is \(f(x) = |x|\).
The absolute value measures the distance from zero without considering the direction. Here are some features of absolute value functions:
  • The function \(f(x) = |x|\) produces a V-shaped graph.
  • All values of \(f(x)\) are non-negative, meaning it鈥檚 zero or positive.
Examining our original function \(f(x) = |x - 2|\), the absolute value means every value of \(x - 2\) is taken as positive. The equation \(f(-x) = |x + 2|\) shows this transformation. Thus comparing both, \(f(x)\) isn鈥檛 equal to \(f(-x)\) or \(-f(x)\). This unique property of absolute value functions helps determine whether they are even or odd based on their symmetry.

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Most popular questions from this chapter

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=1+\frac{1}{x^{2}}$$

Use transformations to graph each function and state the domain and range. $$y=-\frac{1}{2}|x|+40$$

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=|x|+1$$

Use transformations to graph each function and state the domain and range. $$y=-\sqrt{x-3}+1$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation. $$\sqrt{x+3}-2 \geq 0$$
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