Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 17-22, sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 0 \\ -1 & \text { if } x>0\end{array} \quad(a=0)\right.\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)\) consists of a line segment with a slope of 1 for \(x \leq 0\) and a horizontal line at -1 for \(x > 0\). The limit of the function as \(x \rightarrow 0\) is \(\lim_{x\rightarrow 0} f(x) = -1\).

Step 1: Identify function domains and their equations

The function is divided into two parts: one valid for \(x \leq 0\) and the other for \(x > 0\). We have the following two equations: 1. For \(x \leq 0\): \(f(x) = x - 1\) 2. For \(x > 0\): \(f(x) = -1\)

Step 2: Sketch the graph piece by piece

Plot the graph for each part of the function, considering their specific domains: 1. For \(x \leq 0\): The equation is \(f(x) = x-1\), a linear function with a slope of 1 and a y-intercept of -1. Plot the function for the domain \(x \leq 0\). 2. For \(x > 0\): The equation is \(f(x) = -1\), a horizontal line passing through the y-coordinate of -1. Plot the function for the domain \(x > 0\). The graph should be continuous, with no jumps or breaks, as long as \(x\) is in the appropriate domain for each piece of the function.

Step 3: Calculate the limit at x=a=0

To find the limit as \(x\) approaches \(a = 0\), evaluate the limit for both sides, considering the continuity of the graph. For \(x \leq 0\), \(f(x) = x - 1\): \(\lim_{x\rightarrow 0^{-}} (x-1) = 0 - 1 = -1\) For \(x > 0\), \(f(x) = -1\): \(\lim_{x\rightarrow 0^{+}} (-1) = -1\) Since both the left and the right limits are equal to -1, the limit as \(x\rightarrow 0\) exists and is: \(\lim_{x\rightarrow 0} f(x) = -1\) In conclusion, the graph of the function is composed by a line segment of the slope 1 for \(x \leq 0\) and a horizontal line at -1 for \(x > 0\). The limit of the function as \(x \rightarrow 0\) is -1.

Key Concepts

Piecewise Functions

Piecewise functions are a type of function built from multiple sub-functions, each applying to a certain interval of the main function's domain. In the given exercise, we see such a function expressed with different rules depending on the value of the variable:

• For values of \(x\) where \(x \leq 0\), the function is defined as \(f(x) = x - 1\).
• For values of \(x\) where \(x > 0\), the function is defined as \(f(x) = -1\).

This kind of setup helps handle complex real-world scenarios where a single formula isn't adequate for all inputs.
When dealing with piecewise functions, pay close attention to the respective intervals to understand how each piece is applied. The behavior of these functions can be vastly different at the boundaries where the rules change. These boundaries are often points of interest, especially when analyzing limits, as we did with \(x \rightarrow 0\). Here, the function switches from one linear behavior to a constant value.

Continuity

Continuity is a property of functions that determines whether small changes in the input \(x\) result in small changes in the output \(f(x)\). For piecewise functions, checking continuity involves looking at each piece and their transition. In mathematical terms, a function \(f(x)\) is continuous at a point \(a\) if:

• \(f(a)\) is defined.
• The limit \(\lim_{x \rightarrow a} f(x)\) exists.
• \(\lim_{x \rightarrow a} f(x) = f(a)\).

In our exercise, as \(x\) approaches 0:- For values approaching from the left \((x \leq 0)\), \(f(x) = x - 1\) approaches -1.- For values approaching from the right \((x > 0)\), \(f(x) = -1\) is already -1.
Both sides agree, so the function is continuous at \(x = 0\) in terms of the limit existing. However, notice that continuity as we approach from either side doesn't guarantee that the same function value is defined at the jump from one piece to another. That situation doesn't necessarily mean the function isn't continuous, as it's often more about the defined purpose and constraints.

Graph Sketching

Graph sketching for piecewise functions involves drawing each piece according to its rule and domain. It's essential to:

• Identify the intervals and their respective equations, such as \(x \leq 0\) and \(x > 0\) from our exercise.
• Draw each part of the function separately before combining them on a single graph.

For the given function:- The segment for \(x \leq 0\) is a straight line with a slope of 1 (like \(y = x - 1\)), starting at the point \((0, -1)\) and extending to negative infinity.- For \(x > 0\), the graph simply remains as a horizontal line at \(y = -1\). This creates a straight section without any slope.
When sketching, you should mark clearly where each piece begins or ends, especially at the transition points. This gives you an insightful visual representation of the function’s behavior. Understanding these sketches can transform how easily you interpret and calculate limits and continuity.