Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array} \quad(a=3)\right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is piecewise-defined as \[f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array}\right.\] After sketching the graph, we find the left-hand limit at \(x=3\) to be \(2\) and the right-hand limit to also be \(2\). Since both limits coincide, the overall limit exists and \(\lim_{x\to 3}f(x)=2\).

Step 1: Understand the function

The function \(f(x)\) is defined piecewise as follows: \[f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array}\right.\] We have two different expressions for \(f(x)\), depending on whether \(x\) is less than or equal to \(3\) or greater than \(3\). Now, let's analyze both expressions separately. Step 2: Sketch the graph for \(x\leq 3\)

Step 2: Sketch the graph for \(x\leq 3\)

For \(x \leq 3\), we have \(f(x) = x - 1\). This is a straight line with a slope of \(1\) and \(y\)-intercept at \(-1\). On the graph, we will draw this line from \(-\infty\) up to, and including, \(x = 3\). Step 3: Sketch the graph for \(x > 3\)

Step 3: Sketch the graph for \(x > 3\)

For \(x > 3\), we have \(f(x) = -2x + 8\). This is a straight line with a slope of \(-2\) and \(y\)-intercept at \(8\). On the graph, we will draw this line for \(x\) greater than \(3\), without including the point \(x = 3\). Step 4: Evaluate the limit

Step 4: Evaluate the limit

Now that we have the sketch of the graph, we need to evaluate the limit of \(f(x)\) as \(x\) approaches \(3\). The limit of the function is dependent on the right-hand limit and the left-hand limit at \(x = 3\). Let's find both these limits: Left-hand limit: \(\lim_{x \rightarrow 3^-} f(x) = \lim_{x \rightarrow 3^-} (x - 1) = 3 - 1 = 2\). Right-hand limit: \(\lim_{x \rightarrow 3^+} f(x) = \lim_{x \rightarrow 3^+} (-2x + 8) = -2(3) + 8 = -6 + 8 = 2\) As both left-hand and right-hand limits are equal, the limit exists and \(\lim_{x \rightarrow 3} f(x) = 2\).

Key Concepts

Piecewise Functions

Piecewise functions are a specific type of function that have different expressions based on different intervals of the independent variable, which in most cases is the variable x. What makes piecewise functions unique is their ability to model situations that change behavior or rules after certain points. For instance, tax brackets, shipping rates, and step functions in electronics are real-world examples where piecewise functions apply.

When confronted with a piecewise function, such as our example \[f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \-2 x+8 & \text { if } x>3\end{array}\right.\], it is important to understand that we are actually dealing with two separate functions melded into one. Each 'piece' of the function is defined over a certain interval. The power and challenge of piecewise functions lie in understanding how these pieces fit together, which often includes identifying any discontinuities or changes in behavior at the boundary points of the intervals.

Graphing Functions

Graphing functions serves as a visual aid to understand the behavior of mathematical equations. When graphing piecewise functions like \[f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \-2 x+8 & \text { if } x>3\end{array}\right.\], each piece is graphed over its respective interval. For the first piece, we graph the line \(x - 1\), which has a slope of 1 and crosses the y-axis at -1 (y-intercept). It's represented on a graph as a line extending indefinitely to the left and ending at the point \(x = 3\).

For the second piece, we graph the line \(-2x + 8\), with a slope of -2 and a y-intercept at 8. It begins just after \(x = 3\) and extends indefinitely to the right. These graphs help us see the overall shape of the piecewise function and how it behaves, especially as it approaches the boundaries defined by the conditions in the piecewise definition.

Limits of Functions

The concept of limits in calculus is fundamental for studying the behavior of functions as they approach a certain value, without necessarily reaching that value. For a limit to exist at a certain point, the value that the function approaches from the left (left-hand limit) must be the same as the value it approaches from the right (right-hand limit).

When evaluating the limit of a piecewise function such as \[f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \-2 x+8 & \text { if } x>3\end{array}\right.\], at \(x = 3\), we must consider both the left-hand limit and right-hand limit individually: \[\lim_{x \rightarrow 3^-} f(x) \text{ and } \lim_{x \rightarrow 3^+} f(x)\]. Since both approach 2, we can say that the overall limit of \(f(x)\) as \(x\) approaches 3 is 2. Graphing the function can often make it much easier to see this, as we can visually inspect the behavior of the function on both sides of the value \(x = 3\).