Problem 16 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ What new value should be assigned to \(f(1)\) to make the new function continuous at \(x=1 ?\)

Step-by-Step Solution

Answer

To make the function \(f(x)\) continuous at \(x=1\), assign \(f(1) = 2\).

Step 1: Understand Continuity

A function \(f(x)\) is said to be continuous at a certain value \(x=a\) if the following three conditions are met: 1. \(f(a)\) is defined. 2. The limit as \(x\) approaches \(a\) exists.3. The limit as \(x\) approaches \(a\) equals \(f(a)\). So to make a function continuous at a certain point, the value of the function at that point should be the same as the limit of the function at that point.

Step 2: Calculate limit at \(x=1\)

From the right side, i.e, for \(1 < x < 2\), the function is given by \(-2x + 4\). Therefore, the limit of \(f(x)\) as \(x\) approaches \(1\) from the right (denoted by \(\lim_{{x \to 1^+}} f(x)\)) equals to \(-2(1) + 4 = 2\).\nFrom the left side, i.e, for \(0 < x < 1\), the function is \(2x\). So, the limit as \(x\) approaches \(1\) from the left (denoted by \(\lim_{{x \to 1^-}} f(x)\)) equals to \(2(1) = 2\).\nSince, the limit at \(x = 1\) from both sides exists and is the same, we say that \(\lim_{{x \to 1}} f(x) = 2\).

Step 3: Assign new value to \(f(1)\) to make function continuous

According to the third criteria of continuity, to make the function \(f(x)\) continuous at \(x=1\), \(f(1)\) should be equal to the limit of \(f(x)\) at \(x=1\), so, to make the function continuous at \(x=1\), the new value that should be assigned to \(f(1)\) is \(\lim_{{x \to 1}} f(x) = 2\).

Key Concepts

Piecewise Functions

Piecewise functions are a special type of function made up of multiple sub-functions, each of which applies to a specific interval on the function's domain. These functions help model scenarios where different rules apply to different situations or inputs
In a piecewise function, you have separate rules or formulas for different sections of the variable's domain. It looks almost like a menu of options: one formula might apply when the input variable is within a certain range, and another formula kicks in once you hit another range.
To read a piecewise function, identify which formula to use based on the value of the input. It is crucial to understand the conditions or intervals for each formula to know exactly how the function behaves at various points on the number line. Often, questions about continuity arise in piecewise functions at the boundaries where the formulas meet, especially if the transition isn't seamless.

Limits

Limits are fundamental in calculus and play a crucial role in understanding the behavior of functions as they approach a certain point. Think of a limit as the value that a function approaches as the input (or 'x' value) approaches some number.
To determine a limit, you observe what's happening to the function's output as the input value gets closer and closer to the point in question. It's perfectly normal for the actual function value not to reach the limit, which is the beauty of limits—they tell us about the function's behavior rather than its exact value at a point.
When working with piecewise functions, especially, checking limits from both sides is vital. This ensures that the function behaves consistently as it approaches the boundary from either side. If both one-sided limits agree, then the general limit at that point exists and is equal to those one-sided limits. Otherwise, the limit doesn't exist.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It provides tools for analyzing changes and is quintessential in physics, engineering, economics, statistics, and much more.
One of the fundamental aspects of calculus is its ability to handle continuous change and describe variable rates. For instance, by using limits, we can define the derivative of a function—this describes how the function value changes as its input changes, whereas integrals can tell us about accumulation or total values
In piecewise and other functions, calculus allows us to analyze continuity and differentiability. It helps us figure out whether there's a smooth transition between pieces of a piecewise function or if a function can be seamlessly modeled or predicted in real-world scenarios. This makes calculus incredibly powerful and widely applicable.