Problem 27 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. The diving board function, defined by $$ f(x)=\left\\{\begin{array}{ll} 0 & \text { for } x<0 \\ 1 & \text { for } x \geq 0 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph is a piecewise constant function with a jump at \( x = 0 \), changing from 0 for \( x < 0 \) to 1 for \( x \geq 0 \).

Step 1: Identify the Function Type

The function provided is a piecewise function, with different expressions for different intervals of the independent variable, \( x \).

Step 2: Examine the Conditions

The function has two conditions based on the value of \( x \): one for \( x < 0 \) and another for \( x \geq 0 \).

Step 3: Determine the Output for \( x < 0 \)

For all \( x < 0 \), the function value is \( f(x) = 0 \). This is represented by a horizontal line along the x-axis, ranging from negative infinity to zero exclusively.

Step 4: Determine the Output for \( x \geq 0 \)

For all \( x \geq 0 \), the function value is \( f(x) = 1 \). This means the graph is a horizontal line at \( y = 1 \) starting at \( x = 0 \) and extending to positive infinity.

Step 5: Identify the Jump at \( x = 0 \)

There is a jump discontinuity at \( x = 0 \), where the function abruptly changes from 0 to 1. The graph has an open circle at \( (0, 0) \) and a closed circle at \( (0, 1) \) to represent this change.

Step 6: Sketch the Graph

Combine the identified characteristics to sketch the graph: draw the horizontal line at \( y = 0 \) for \( x < 0 \) and another horizontal line at \( y = 1 \) for \( x \geq 0 \). Indicate the open and closed circles at \( (0, 0) \) and \( (0, 1) \), respectively.

Key Concepts

Function Graphing

Graphing a function is like telling a visual story about how a function behaves. When we graph a piecewise function, like our diving board function, we need to pay attention to how the function is defined in different parts of the domain.

Each part of the function has its own rule. For the diving board function:

• When \( x < 0 \), we follow the rule \( f(x) = 0 \).
• When \( x \geq 0 \), the rule is \( f(x) = 1 \).

This means for \( x < 0 \), the graph will be a horizontal line along the x-axis, which is the line where all points have a \( y \)-coordinate of zero. For \( x \geq 0 \), the graph jumps to a horizontal line at \( y = 1 \).

In sketching such a function, you'll need:

• To decide where each part of the function starts and stops.
• To show clearly where the graph jumps by using symbols like open and closed dots.

Jump Discontinuity

Imagine you're watching a line moving smoothly when, suddenly, it makes a jump from one point to another. That's what a jump discontinuity is.

In our piecewise function, the jump discontinuity happens at \( x = 0 \). Here, the function changes abruptly from \( f(x) = 0 \) for \( x < 0 \) to \( f(x) = 1 \) for \( x \geq 0 \). This jump makes a clear "break" in the graph.

To highlight a jump discontinuity on the graph, we use:

• An open circle at the point where the line would have continued if not for the jump. Here, it's at \( (0, 0) \), suggesting the function does not include this point when \( x = 0 \).
• A closed circle where the line actually jumps to, which is at \( (0, 1) \), meaning this point is included.

Step Function

Step functions are like staircases in math. Each "step" shows a constant value over a certain interval.

Our diving board function resembles a step function with just two steps:

• The step at \( y = 0 \) for \( x < 0 \)
• The step at \( y = 1 \) for \( x \geq 0 \)

In this step function, each step is defined clearly by the function's rules. There's no smooth transition between these steps, making them clearly distinguished from each other.

This is a key feature of step functions, especially in cases where the graph needs to represent a scenario with distinct intervals, like our function which "jumps" from 0 to 1 at a specific point in its domain.