Question

Determine whether the given function is even, odd, or neither. $$\tan 2x$$

Short answer

Answer: The function $f(x)=\tan 2x$ is an odd function.

Step-by-step solution

Write down the function

The given function is $$f(x)=\tan 2x$$

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: $$f(-x)=-\tan (2x)$$

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.

Conclusion

Since $f(-x)=-f(x)$, the given function, $f(x)=\tan 2x$, is an odd function.

Key concepts

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.

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.

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.

Reflection

One common transformation is reflection, which flips the graph over a certain axis. This is particularly relevant when discussing even and odd functions. As observed in the exercise, the negative in the expression $-\text{tan}(2x)$ indicates a reflection across the x-axis, establishing that the function is odd. For even functions, a reflection over the y-axis doesn't change the function.

Scaling and Shifting

Other transformations include scaling, which changes the size of the graph, and shifting, which moves the graph horizontally or vertically without altering its shape. In the context of $\text{tan}(2x)$ from our exercise, the factor of '2' inside the tangent function represents a horizontal compression by a factor of 1/2, a concept that is beneficial for students to understand in conjunction with function parity.