Problem 16 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(t)=|t|+t $$

Step-by-Step Solution

Answer

The graph is a line y=0 for t<0, and a line y=2t for t≥0, meeting at (0,0).

Step 1: Understand the Function

The function given is \( f(t) = |t| + t \). This is a piecewise function, combining an absolute value, \(|t|\), with a linear component, \(t\). To sketch it, we need to consider how these two components interact for different values of \(t\).

Step 2: Break Down the Absolute Value

The absolute value \(|t|\) affects the function differently depending on \(t\). For \(t \geq 0\), \(|t| = t\), and for \(t < 0\), \(|t| = -t\). This means the function \(f(t)\) can be rewritten piecewise as: \[ f(t) = \begin{cases} 2t, & \text{if } t \geq 0 \ 0, & \text{if } t < 0 \end{cases}\]

Step 3: Identify Key Points

Consider specific values to get crucial points of the graph. For instance, at \(t = 0\), \(f(0) = 0\). For \(t > 0\), the function simplifies to \(f(t) = 2t\), so \(f(1) = 2\), \(f(2) = 4\), etc. For \(t < 0\), the function simplifies to \(f(t) = 0\), so \(f(-1) = 0\), \(f(-2) = 0\).

Step 4: Sketch the Graph

Plot the points identified: start from \((0,0)\), which is a turning point. For \(t \geq 0\), draw a line with a slope of 2 passing through \((0,0)\) upwards. For \(t < 0\), the function is constant and equal to zero, forming a horizontal line at \(y=0\).

Step 5: Analyze the Graph

Observe that the graph is a line running along the x-axis for negative \(t\) and rises with a slope of +2 for non-negative \(t\). The function has a corner at \(t=0\), where the slope changes abruptly.

Key Concepts

Absolute Value Function

An absolute value function is a type of piecewise function that affects the input, or the "argument," by making sure its output value is non-negative. The absolute value of a number \(|t|\) represents its distance from zero on a number line, without considering its direction. It effectively "flips" any negative values into positive ones. For the function we are considering, \(|t|\), it works as follows:

• For \(t \geq 0\), \(|t| = t\), because positive numbers remain unchanged by the absolute value.
• For \(t < 0\), \(|t| = -t\), because negative numbers become positive when they are flipped by the absolute value operation.

In our function \(f(t) = |t| + t\), understanding how \(|t|\) works is crucial because it affects the behavior of the piecewise function explained below.

Piecewise Function

Piecewise functions are functions that have different expressions based on the input value; they are defined in 'pieces' using multiple sub-functions. A great way to understand piecewise functions is to break them down according to the conditions given for the input variable. For example, in our function \(f(t) = |t| + t\), we derived it into a piecewise representation:

• For \(t \geq 0\), the function becomes \(f(t) = 2t\) because \(|t| + t = t + t = 2t\).
• For \(t < 0\), the function simplifies to \(f(t) = 0\) because \(|t| + t = (-t) + t = 0\).

This separation allows us to sketch the graph of the function easily by understanding that each piece behaves differently depending on the domain it represents. This way, sketching can be more organized, visualizing one piece at a time based on specific input intervals.

Linear Function

A linear function is a function whose graph is a straight line. It has a general form of \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. In the simplified parts of our piecewise function, the segment \(f(t) = 2t\) is a linear function:

• The slope, \(m=2\), indicates that for each horizontal step (increase in \(t\)), the function's value goes up by \(2\) units vertically. This steepness of the line is visually observable.
• The y-intercept is \(b=0\), meaning the line crosses the origin \((0,0)\). This occurs because no constant term is added in the equation \(2t\).

On the graph, this corresponds to a line with a slope of \(+2\) for all non-negative \(t\). Recognizing the linear part helps in sketching and understanding these segments effortlessly as part of the overall function.