Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(t)=\frac{t(3+t)}{t}\). (a) Sketch the graph of \(f(t)\). (b) What are the domain and range of \(f(t) ?\)

Step-by-Step Solution

Answer

The graph of \(f(t)\) is a straight line \(f(t) = 3+t\) with a hole at \(t=0\), and its domain is \(-\infty < t < \infty\), \(t \neq 0\) with range \(-\infty < f(t) < \infty\).

Step 1: Simplifying the function

Simplify the given function \(f(t)=\frac{t(3+t)}{t}\). Cancel out the numerator and denominator 't' which gives \(f(t)=3+t\).

Step 2: Sketching the graph

To sketch the graph of this linear function, plot a few points, for example take \(t=-3,-2,-1,0,1,2,3\). Substitute all these values into \(f(t) = 3+t\), plot them and connect the dots. Remember, as \(f(t)\) is undefined at \(t=0\), we should indicate that by making a hole at \(t=0\) on the graph.

Step 3: Identifying the Domain

The domain of a function is the set of all real numbers that \(t\) can take. In this case, \(t\) can take any real number except zero, because at zero the original function is undefined. So the domain is \(-\infty < t < \infty\), \(t \neq 0\).

Step 4: Finding the Range

The range is the set of all real numbers that \(f(t)\) can take. As the function \(f(t) = 3+t\) is a linear function its range is all real numbers. Therefore, the range is \(-\infty < f(t) < \infty\).

Key Concepts

Function Simplification

Function simplification is a crucial step in making a complex expression much easier to understand and work with. In this exercise, we are simplifying the function \(f(t)=\frac{t(3+t)}{t}\). Simplification often involves canceling out common factors. Here, since the numerator and the denominator share the factor \(t\), we can simplify the function to \(f(t) = 3+t\).

• Canceling like terms helps reduce complexity.
• A simplified function is often easier to analyze.

Remember though, while simplifying, do not overlook other important aspects such as undefined points, which we'll address in later sections.

Graph Sketching

Graph sketching is an effective way to visualize the behavior of a function. After simplifying \(f(t)\) to \(3 + t\), we note that this describes a straight line, or a linear function. Here's how to proceed:

• Choose several values for \(t\) (e.g., \(-3, -2, -1, 0, 1, 2, 3\)).
• Calculate corresponding \(f(t)\) values for each chosen \(t\). This forms a series of coordinates.
• Plot these coordinates on a graph.
• Connect the points to reveal the line.

Be aware that there's a peculiarity. Since the original function is undefined at \(t = 0\), you need to place a small circle or a break in the line at \(t = 0\) to indicate the hole in the graph.

Linear Function

A linear function, like the one we have here, is represented by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In \(f(t) = 3 + t\), the slope is 1, which means the line rises one unit vertically for every unit it moves horizontally. The y-intercept here is 3, indicating that the line crosses the y-axis at \(f(t) = 3\).

• Linear functions make straight lines when graphed.
• The slope determines the direction and steepness of the line.
• The y-intercept is where the line crosses the y-axis.

Understanding these components makes it easier to analyze and sketch the graph accurately.

Undefined Points

An undefined point in a function occurs when the function does not have a valid output for a given input. For our function \(f(t) = \frac{t(3+t)}{t}\), it is undefined at \(t = 0\).

• Dividing by zero leads to undefined behavior, as seen in the original form of \(f(t)\).
• Even after simplification, the undefined nature at \(t = 0\) remains. It is crucial to remember this, despite the simplified form not explicitly showing it.

On a graph, such undefined points are represented by open circles or holes to indicate where \(f(t)\) does not exist. This concept is important for complete and accurate graph sketching.