Problem 73
Question
Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} 2-x & \text { if } x \leq 4 \\ x-6 & \text { if } x>4 \end{array}\right.\\\ \text { [Hint: See Exercise } 39 .] \end{array} $$
Step-by-Step Solution
Verified Answer
The function \(f(x)\) is continuous everywhere.
1Step 1: Identify Pieces of the Function
The given function is a piecewise function consisting of two parts: 1. For \(x \leq 4\), the function is \(f(x) = 2 - x\).2. For \(x > 4\), the function is \(f(x) = x - 6\).
2Step 2: Determine Continuity at Different Sections
To determine continuity, we examine each piece of the function separately. Both expressions \(2-x\) and \(x-6\) are polynomials, which are continuous for all their domain values. Thus, each part of the function is continuous on its respective domain.
3Step 3: Check Continuity at the Transition Point
To determine continuity at the transition point \(x = 4\), we need to check the following three conditions for continuity at a point:- \(f(4)\) should exist.- \(\lim_{{x \to 4^-}} f(x) = \lim_{{x \to 4^+}} f(x)\).- The common limit is equal to \(f(4)\).Let's verify these:1. \(f(4) = 2 - 4 = -2\).2. \(\lim_{{x \to 4^-}} f(x) = 2 - 4 = -2\).3. \(\lim_{{x \to 4^+}} f(x) = 4 - 6 = -2\).
4Step 4: Conclusion on Continuity
Since both the left-hand limit \(\lim_{{x \to 4^-}} f(x)\) and the right-hand limit \(\lim_{{x \to 4^+}} f(x)\) are \(-2\), and these limits equal \(f(4)\), the function is continuous at \(x = 4\). Therefore, \(f(x)\) is continuous everywhere.
Key Concepts
ContinuityPolynomial FunctionsLimits
Continuity
Continuity is a fundamental concept in calculus, describing a function that has no breaks, jumps, or holes in its graph. A function is considered continuous at a point if three conditions are met:
- The function is defined at that point.
- The limit of the function as it approaches the point from both sides exists.
- The limit is equal to the function value at that point.
When analyzing a piecewise function like the one in the exercise, we must check continuity separately for each piece and also at the transition points where the definition of the function changes.
For our function, the transition occurs at the point \(x = 4\). We determine continuity by checking the value of \(f(x)\), its limit as \(x\) approaches from the left \((\lim_{{x \to 4^-}} f(x))\) and from the right \((\lim_{{x \to 4^+}} f(x))\), to see if they are all the same. If they match, the function is continuous at that point.
- The function is defined at that point.
- The limit of the function as it approaches the point from both sides exists.
- The limit is equal to the function value at that point.
When analyzing a piecewise function like the one in the exercise, we must check continuity separately for each piece and also at the transition points where the definition of the function changes.
For our function, the transition occurs at the point \(x = 4\). We determine continuity by checking the value of \(f(x)\), its limit as \(x\) approaches from the left \((\lim_{{x \to 4^-}} f(x))\) and from the right \((\lim_{{x \to 4^+}} f(x))\), to see if they are all the same. If they match, the function is continuous at that point.
Polynomial Functions
Polynomial functions are algebraic expressions consisting of terms with non-negative integer exponents. They take the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) where the \(a_i\) are constants. Polynomials like \(2-x\) and \(x-6\) are often used for modeling because they are continuous and smooth over their entire domain.
Polynomials are defined on all real numbers, and their graphs are continuous everywhere, meaning you can draw their graphs without lifting your pencil. This characteristic is crucial when considering piecewise functions, as each segment defined by a polynomial is guaranteed to be continuous on its own.
In our piecewise function, each segment is linear, a simple case of a polynomial. Therefore, \(2-x\) and \(x-6\) are continuous for any value defined by their respective domains.
Polynomials are defined on all real numbers, and their graphs are continuous everywhere, meaning you can draw their graphs without lifting your pencil. This characteristic is crucial when considering piecewise functions, as each segment defined by a polynomial is guaranteed to be continuous on its own.
In our piecewise function, each segment is linear, a simple case of a polynomial. Therefore, \(2-x\) and \(x-6\) are continuous for any value defined by their respective domains.
Limits
Limits help in understanding the behavior of a function as it approaches a particular point or as it extends towards infinity. They are foundational in calculus, allowing for the analysis of function behavior at moments of uncertainty, such as at the transition points of piecewise functions.
When evaluating the limit of a piecewise function at its transition point \(x\), you calculate the limit from the left \((\lim_{{x \to x^-}} f(x))\) and the limit from the right \((\lim_{{x \to x^+}} f(x))\). For continuity, these two limits must be equal, and also match the value of the function at that point.
In the original exercise, we computed \(\lim_{{x \to 4^-}} f(x) = -2\) and \(\lim_{{x \to 4^+}} f(x) = -2\). Since these limits match and are equal to \(f(4) = -2\), it confirms the function remains continuous across \(x = 4\). Understanding limits and their calculation ensure clarity in assessing the continuity of piecewise functions.
When evaluating the limit of a piecewise function at its transition point \(x\), you calculate the limit from the left \((\lim_{{x \to x^-}} f(x))\) and the limit from the right \((\lim_{{x \to x^+}} f(x))\). For continuity, these two limits must be equal, and also match the value of the function at that point.
In the original exercise, we computed \(\lim_{{x \to 4^-}} f(x) = -2\) and \(\lim_{{x \to 4^+}} f(x) = -2\). Since these limits match and are equal to \(f(4) = -2\), it confirms the function remains continuous across \(x = 4\). Understanding limits and their calculation ensure clarity in assessing the continuity of piecewise functions.
Other exercises in this chapter
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