Problem 57 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=1+\tan ^{2} x$$.

Step-by-Step Solution

Answer

The domain of the function \(f(x) = 1 + \tan^2 x\) is \(x \neq \frac{\pi}{2} + k\pi\) where \(k\) is an integer, and the range is \([1,\infty)\).

Step 1: Evaluate the function

The given function is \(f(x)=1+\tan^{2}x\). This is a sum of 1 and the square of the tangent function of \(x\). Since squared values are always non-negative (\( \geq 0\)) and we are adding 1, this expression will always be equal to or greater than 1. Therefore the range of \(f\) is \( [1, \infty) \).

Step 2: Determine the domain

Next, we need to identify the domain, that is, the values of \(x\) for which the function is defined. As noted, the tangent function is undefined at \(\frac{\pi}{2}+k\pi\), where \(k\) is an integer, because we would be dividing by zero at these points. Therefore, the domain of \(f\) is all real numbers except these points, which we can express as \(x \neq \frac{\pi}{2} + k\pi\) where \(k\) is an integer.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially when dealing with angles and periodic phenomena. These functions include sine, cosine, tangent, and more. They relate the angles of a triangle to the lengths of its sides. In many math problems, trigonometric functions help in modeling waves, oscillations, and other cyclical processes.

• Sine (\( \sin \theta \)): Represents the opposite side over the hypotenuse.
• Cosine (\( \cos \theta \)): Represents the adjacent side over the hypotenuse.
• Tangent (\( \tan \theta \)): Represents the opposite side over the adjacent side.

These functions are periodic, meaning they repeat their values in regular intervals. Understanding these functions is key to solving problems involving cycles and waves.

Exploring the Tangent Function

The tangent function, often written as \( \tan x \), is a trigonometric function that represents the ratio of the sine to the cosine of an angle. It is particularly interesting because it has distinct properties, such as being undefined at certain points.

• The tangent function is undefined when the cosine of the angle is zero.
• This occurs at angles \( \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

At these points, the function approaches positive or negative infinity, which results in vertical asymptotes on a graph. Graphically, the tangent function repeats every \( \pi \) radians, and this periodic nature is crucial in mathematics.

Defining the Domain of a Function

The domain of a function refers to all the possible input values (usually represented by \( x \)) that make the function work without breaking any mathematical rules. For trigonometric functions like tangent, determining the domain involves understanding where they are undefined.

• The domain of \( \tan x \) excludes points where it is undefined, such as \( x eq \frac{\pi}{2} + k\pi \).
• This means our original function \( f(x) = 1 + \tan^2 x \) is defined everywhere except these undefined points.

By removing these specific points from consideration, we ensure that our function \( f(x) \) remains well-defined and can be calculated correctly.

Identifying the Range of a Function

The range of a function is the set of all possible output values (commonly represented by \( f(x) \)) that result from using the domain values. For our given function \( f(x) = 1 + \tan^2 x \), understanding the characteristics of the tangent function helps identify the range.

• The square of any real number, including tangent, is always non-negative, meaning \( \tan^2 x \geq 0 \).
• Adding 1 to \( \tan^2 x \) results in \( f(x) \geq 1 \).

Thus, the range of this function is all real numbers greater than or equal to 1, expressed as \([1, \infty)\). This understanding is essential for solving various mathematical problems accurately.