Chapter 10: Problem 3
Determine whether the given function is even, odd, or neither. $$ \tan 2 x $$
Short Answer
Expert verified
Answer: The function \(f(x) = \tan 2x\) is an odd function.
Step by step solution
01
Write down the function
The given function is
$$
f(x) = \tan 2x
$$
02
Find the value of \(f(-x)\)
To find \(f(-x)\), replace \(x\) with \(-x\) in the given function:
$$
f(-x) = \tan 2(-x)
$$
Using the property \(\tan(-\theta) = -\tan(\theta)\), we have:
$$
f(-x) = -\tan(2x)
$$
03
Compare \(f(-x)\) and \(f(x)\)
Now we have:
$$
f(x) = \tan 2x
$$
and
$$
f(-x) = -\tan 2x
$$
This clearly shows that \(f(-x) = -f(x)\), which is the condition for an odd function.
04
Conclusion
Since \(f(-x) = -f(x)\), the given function, \(f(x) = \tan 2x\), is an odd function.
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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 the concepts of even and odd functions provides insight into symmetrical properties of functions, which is essential in fields such as engineering and physics. A function is considered even if for every input value x, the condition \( f(x) = f(-x) \) holds true. This indicates symmetric behavior around the y-axis. Examples include functions like \( f(x) = x^2 \) and \( f(x) = \text{cos}(x) \).
Conversely, a function is deemed odd if for all input values x, the condition \( f(-x) = -f(x) \) is satisfied, showing symmetry with respect to the origin. Functions like \( f(x) = x^3 \) and \( f(x) = \text{sin}(x) \) illustrate this property. In the given exercise, the function \( f(x) = \text{tan}(2x) \) is identified as odd because it meets the criteria of \( f(-x) = -f(x) \) when the input is negated.
Conversely, a function is deemed odd if for all input values x, the condition \( f(-x) = -f(x) \) is satisfied, showing symmetry with respect to the origin. Functions like \( f(x) = x^3 \) and \( f(x) = \text{sin}(x) \) illustrate this property. In the given exercise, the function \( f(x) = \text{tan}(2x) \) is identified as odd because it meets the criteria of \( f(-x) = -f(x) \) when the input is negated.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They play a crucial role in simplifying expressions and solving trigonometric equations. Some fundamental identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities.
For example, the identity \( \text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \) can be used to relate the tangent function to the sine and cosine functions. In our exercise, to decide if the function \( f(x) = \text{tan}(2x) \) is odd, we rely on the identity \( \text{tan}(-\theta) = -\text{tan}(\theta) \) to evaluate the expression for \( f(-x) \) and confirm the function's parity.
For example, the identity \( \text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \) can be used to relate the tangent function to the sine and cosine functions. In our exercise, to decide if the function \( f(x) = \text{tan}(2x) \) is odd, we rely on the identity \( \text{tan}(-\theta) = -\text{tan}(\theta) \) to evaluate the expression for \( f(-x) \) and confirm the function's parity.
Function Transformation
Function transformation is a method of modifying a function's graph through various operations such as shifting, reflecting, stretching, or compressing. It is a powerful tool in analyzing the behavior of functions and in understanding how different algebraic modifications impact the graph's shape and position.