Question

Find \(D_{f}\) (a) \(f(x)=\tan x\) (b) \(f(x)=\frac{1}{\sqrt{1-|\sin x|}}\) (c) \(f(x)=\frac{1}{x(x-1)}\) (d) \(f(x)=\frac{\sin x}{x}\) (e) \(e^{[f(x)]^{2}}=x, \quad f(x) \geq 0\)

Short answer

(a) \(f(x)=\tan x\) Domain: \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq \frac{(2n+1)\pi}{2},\ n \in \mathbb{Z} \}\) (b) \(f(x)=\frac{1}{\sqrt{1-|\sin x|}}\) Domain: \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq n\pi +(-1)^{n}\frac{\pi}{2},\ n \in \mathbb{Z}\}\) (c) \(f(x)=\frac{1}{x(x-1)}\) Domain: \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq 0, x \neq 1\}\) (d) \(f(x)=\frac{\sin x}{x}\) Domain: \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq 0\}\) (e) \(e^{[f(x)]^{2}}=x,\quad f(x) \geq 0\) Domain: \(D_{f} = \{x \in \mathbb{R} \ |\ x > 0\}\)

Step-by-step solution

Identify points of discontinuity

The tangent function has vertical asymptotes where the cosine function is zero, i.e., at every odd multiple of \(\frac{\pi}{2}\). It makes the function undefined at these points.

Write the domain

The domain of \(f(x)=\tan x\) is all real numbers except the odd multiples of \(\frac{\pi}{2}\). So, \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq \frac{(2n+1)\pi}{2},\ n \in \mathbb{Z} \}\) (b) \(f(x)=\frac{1}{\sqrt{1-|\sin x|}}\)

Identify points of discontinuity

The function will be undefined where the denominator is zero or negative, i.e., \(\sqrt{1-|\sin x|} \leq 0\)

Find the values of x where the function is undefined

The denominator is zero when \(|\sin x| = 1\), which occurs when \(x=n\pi +(-1)^{n}\frac{\pi}{2}\) for any integers \(n\). It is not negative since the maximum value of \(|\sin x|\) is 1.

Write the domain

The domain of \(f(x)\) is all real numbers except the values where the function is undefined. So, \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq n\pi +(-1)^{n}\frac{\pi}{2},\ n \in \mathbb{Z}\}\) (c) \(f(x)=\frac{1}{x(x-1)}\)

Identify points of discontinuity

The function will be undefined when the denominator is zero, which occurs when \(x = 0\) or \(x = 1\)

Write the domain

The domain of \(f(x)\) is all real numbers except the values where the function is undefined. So, \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq 0, x \neq 1\}\) (d) \(f(x)=\frac{\sin x}{x}\)

Identify points of discontinuity

The function will be undefined when the denominator is zero, which occurs when \(x = 0\)

Write the domain

The domain of \(f(x)\) is all real numbers except the values at which the function is undefined. So, \(D_{f} = \{x \in \mathbb{R} \ |\ x \neq 0\}\) (e) \(e^{[f(x)]^{2}}=x, \quad f(x) \geq 0\)

Identify the restrictions on x

The given condition dictates that \(f(x) \geq 0\) and the base of the exponential function is positive. Therefore, \(x\) must be positive for any value of \(f(x)\).

Write the domain

The domain of \(f(x)\) is all real numbers that satisfy \(x > 0\). So, \(D_{f} = \{x \in \mathbb{R} \ |\ x > 0\}\)

Key concepts

Discontinuity Points

Discontinuity points are specific values in the domain of a function at which the function is not continuous. A function is continuous at a point if the function is defined there, and if the limit of the function as it approaches the point from both directions exists and is equal to the function's value at that point. When discussing functions like
\(f(x) = \frac{1}{\sqrt{1-|\sin x|}}\), we look for where the function's denominator is zero since division by zero is undefined. Discontinuity can also arise when there's an asymptote; for example, the tangent function
\(f(x) = \tan x\) has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\), which are precisely its discontinuity points. Identifying discontinuity points is a crucial step in determining the function's domain, as these points are excluded from it.

Real Analysis

Real Analysis is a field of mathematics that deals with the properties of real numbers, sequences, series, and real-valued functions. It provides the rigorous foundations for calculus and deals extensively with the limits and continuity of functions. In performing real analysis, we deal with issues like convergence, the construction of real numbers, and the precise definitions of derivatives and integrals. For example, when analyzing the domain of functions like \(f(x) = \frac{1}{x(x-1)}\), real analysis provides the tools we need to prove that the domain must exclude the points where the function's denominator is zero, since these points lead to discontinuities in the real number system.

Function Domain Determination

The domain of a function consists of all the input values (x-values) for which the function is defined. Function domain determination is the process of figuring out which values of 'x' can be plugged into the function without causing issues like division by zero, taking the square root of a negative number, or any other undefined operations. Take the function \(f(x) = \frac{\sin x}{x}\). The domain determination here necessitates excluding the point where the denominator is zero, i.e., \(x = 0\), ensuring the function's definition over its domain. Thus, the domain is all real numbers except where the function presents these undefined operations.

Asymptotes in Functions

An asymptote is a line that a function approaches but never actually reaches, no matter how far out its graph extends. There are different types of asymptotes: vertical, horizontal, and oblique. Vertical asymptotes typically occur in rational functions where the denominator can be zero--for instance, \(f(x) = \tan x\) has vertical asymptotes because the cosine function in the denominator equals zero at odd multiples of \(\frac{\pi}{2}\). Horizontal asymptotes indicate the behavior of a function as 'x' heads towards positive or negative infinity and reflects a function's end-behavior. Understanding asymptotes' location and behavior is fundamental in sketching graphs of functions and determining their limits at different points.