Domain Of F(x) = 3tan(3x + Π/6) + 1 Explained
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of trigonometric functions, specifically focusing on the tangent function and its domain. We'll be dissecting the function f(x) = 3tan(3x + π/6) + 1, a seemingly simple expression that holds a wealth of mathematical concepts. Understanding the domain of a function is crucial in mathematics as it defines the set of all possible input values (x-values) for which the function produces a valid output. So, buckle up, and let's embark on this mathematical journey together!
Understanding the Tangent Function and Its Domain
Before we jump into the specifics of our function f(x), let's take a step back and refresh our understanding of the tangent function itself. The tangent function, often written as tan(x), is one of the fundamental trigonometric functions. It's defined as the ratio of the sine function to the cosine function: tan(x) = sin(x) / cos(x). This definition is key to understanding why the tangent function has a unique and interesting domain.
The domain of a function is the set of all possible input values (x-values) that will produce a valid output. Now, considering the tangent function's definition as a ratio, we immediately encounter a potential issue: division by zero. Remember, division by zero is undefined in mathematics. Therefore, the tangent function will be undefined whenever the cosine function in the denominator equals zero. To pinpoint these problematic points, we need to identify where cos(x) = 0.
The cosine function equals zero at angles of π/2 + nπ, where n is any integer. This means that the graph of the cosine function crosses the x-axis (where the function value is zero) at π/2, 3π/2, -π/2, and so on. Consequently, the tangent function is undefined at these same angles. This is because the denominator in the tan(x) = sin(x) / cos(x) expression becomes zero, leading to an undefined result.
Visually, this translates to vertical asymptotes on the graph of the tangent function. Asymptotes are imaginary vertical lines that the function approaches but never quite reaches. At the points where the tangent function is undefined, the graph shoots off towards positive or negative infinity, creating these characteristic vertical asymptotes. The tangent function repeats its pattern between these asymptotes, resulting in a periodic graph with a period of π. This periodic nature arises from the periodic behavior of both the sine and cosine functions that define it.
Therefore, the domain of the basic tangent function, tan(x), is all real numbers except for x = π/2 + nπ, where n is an integer. This exclusion is crucial for maintaining a well-defined function and avoiding undefined results. This understanding of the basic tangent function is the foundation upon which we'll build our analysis of the more complex function f(x) = 3tan(3x + π/6) + 1.
Deciphering the Domain of f(x) = 3tan(3x + π/6) + 1
Now that we have a solid grasp of the basic tangent function's domain, let's tackle the slightly more intricate function, f(x) = 3tan(3x + π/6) + 1. This function is a transformation of the basic tangent function, incorporating a horizontal stretch/compression, a horizontal shift, and a vertical stretch and shift. To find its domain, we need to carefully consider how these transformations affect the original tangent function's points of discontinuity.
The core of this function is the tan(3x + π/6) term. The constants inside the tangent function, namely the 3 and the π/6, are the key players in determining the domain. Remember that the tangent function is undefined when its argument (the expression inside the parentheses) is equal to π/2 + nπ, where n is an integer. This is because, as we discussed earlier, the cosine of these angles is zero, leading to division by zero in the tangent's definition.
So, to find the values of x that make our function undefined, we need to solve the equation: 3x + π/6 = π/2 + nπ. This equation essentially identifies the values of x that cause the argument of the tangent function to land on one of those problematic angles where the cosine is zero. Let's break down the steps to solve this equation:
- Isolate the term with x: Subtract π/6 from both sides of the equation: 3x = π/2 - π/6 + nπ. Simplifying the right side, we get 3x = π/3 + nπ.
- Solve for x: Divide both sides of the equation by 3: x = π/9 + nπ/3. This is a crucial result! It tells us the values of x that must be excluded from the domain of f(x).
This result, x = π/9 + nπ/3, represents a family of values, one for each integer n. When x takes on any of these values, the argument of the tangent function, 3x + π/6, becomes equal to π/2 + nπ, making the tangent function undefined and consequently, making f(x) undefined. Therefore, these values must be excluded from the domain.
The 3 multiplying the x inside the tangent function compresses the graph horizontally, which means the vertical asymptotes are closer together compared to the basic tan(x) function. The π/6 term shifts the graph horizontally. The 3 multiplying the entire tangent function stretches the graph vertically, and the +1 shifts the graph vertically upwards. However, these vertical transformations do not affect the domain, which is solely determined by the horizontal transformations and the inherent discontinuities of the tangent function.
Expressing the Domain Correctly
Now that we've pinpointed the values of x that are not in the domain, we need to express the domain clearly and concisely. The domain of f(x) = 3tan(3x + π/6) + 1 is the set of all real numbers x such that x ≠ π/9 + nπ/3, where n is an integer. This is a precise and mathematically sound way to define the domain. It captures the fact that the function is defined for all real numbers except for the infinite set of points where the tangent function becomes undefined.
Let's consider how this answer relates to the multiple-choice options often presented in mathematical problems. You might see options like:
A. x ≠ π/6 + nπ/6 B. x ≠ π/9 + nπ C. x ≠ ...
By carefully working through the steps we outlined above, we arrived at the correct answer, which might not always be immediately obvious. The key is to remember the fundamental definition of the tangent function and how transformations affect its domain. In our case, the correct answer would be the one that matches our derived domain: x ≠ π/9 + nπ/3.
It's worth noting that the form of the answer might be slightly different but still equivalent. For instance, x ≠ π/9 + nπ/3 can be rewritten as x ≠ (π + 3nπ)/9, which simplifies to x ≠ π(1 + 3n)/9. While the appearance is different, both expressions represent the same set of excluded values.
Visualizing the Domain and Asymptotes
To further solidify our understanding, let's visualize the domain and asymptotes of f(x) = 3tan(3x + π/6) + 1. Imagine graphing the function. You would see a series of tangent function-like curves, but they are compressed horizontally and shifted both horizontally and vertically. The key features to focus on are the vertical asymptotes. These asymptotes occur at the values of x we excluded from the domain: x = π/9 + nπ/3.
If you were to plot these asymptotes on a graph, you'd see them spaced at regular intervals. The distance between consecutive asymptotes represents the