Problem 50 - College Algebra

Question

The domain of each piecewise function is \((-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{rll}4 & \text { if } & x \leq-1 \\\\-4 & \text { if } & x>-1\end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the function consists of two horizontal lines: one at y = 4 for all \(x \leq -1\) and one at y = -4 for all \(x > -1\). The range of the function is {4, -4}.

Step 1: Plot the Function for \(x \leq -1\)

First, let's plot the graph of the function when \(x \leq -1\). Since the function is defined as f(x) = 4 for all \(x \leq -1\), this means that regardless of the specific 'x' value we select, as long as 'x' is less than or equal to -1, the function will always equal 4. We then plot a horizontal line at y = 4 that extends towards the left until it reaches the point at x = -1. Due to \(x \leq -1\) condition, you would normally place a closed dot at the point where \(x = -1\), which signifies that x = -1 is included in that part of the function.

Step 2: Plot the Function for \(x > -1\)

Now let's plot the graph of the function when \(x > -1\). As the function is defined as f(x) = -4 for all \(x > -1\), regardless of the specific 'x' value, as long as 'x' is greater than -1, the function will always be -4. We then plot a horizontal line at y = -4 that extends towards the right from the point where x = -1. Because of the condition \(x > -1\), place an open dot at the point where \(x = -1\), signifying that x = -1 is not included in this part of the function.

Step 3: Determine the Range

The range of a function is the set of all possible 'y' values. When looking at our graph, we only see two 'y' values: 4 and -4. Thus, the range of our function is {4, -4}.

Key Concepts

Function Range

The range of a function is all about understanding which values of "y" can emerge from plugging" x" into the function. For piecewise functions like the one in our exercise, this means looking at all the separate pieces of the function.
In our example, the function is split into two parts. First, when \(x \leq -1\), the function is always equal to 4. This tells us that 4 is a piece of our function range. Next, when \(x > -1\), the function equals -4. So -4 makes up the other piece of the range.
By examining the graph, we see only two distinct horizontal lines: one at \(y = 4\) and the other at \(y = -4\).

• This simple examination shows that the range only consists of these two y-values, \(\{4, -4\}\).
• For any "x" you substitute in, the result will either be 4 or -4, never anything else.

This is a key characteristic of piecewise-defined functions, making them easy to comprehend once you break them down into their components.

Plotting Graphs

To effectively graph a piecewise function like this, it helps to tackle one segment at a time. Here’s a straightforward guide for handling these kinds of functions:
First, identify the separate parts of the function and their respective conditions. In our case:

• Part one is \(f(x) = 4\) for \(x \leq -1\).
• Part two is \(f(x) = -4\) for \(x > -1\).

Next, draw them on a coordinate system. You’ll find each part of the function corresponds to a horizontal line:

• The line \(y = 4\) covers all the points where \(x\) is less than or equal to -1.
• The line \(y = -4\) takes over where \(x\) is greater than -1.

This function showcases a basic two-part graph. Each line can start from a point on the \(x\)-axis, paying attention to whether the point is included (closed dot) or excluded (open dot), like at \(x = -1\). Little details like where to put closed or open dots help define these pieces accurately.

Domain of Function

A function's domain is all about identifying the possible values that "x" can assume. It's the set of all input values for which the function gives a result. In our exercise, this piecewise function has a domain of \((-\infty, \infty)\), meaning any real number can be inserted into it.
Essentially, the domain tells us about the limitations or extent of "x" for which the function is defined.

• Because this function is defined for every real number "x"—due to it covering all values with its pieced-together nature—it ends up having an unlimited domain.
• Each section of the piecewise function accommodates either side or all of this infinite domain, respecting the defined inequalities \((x \leq -1)\) and \((x > -1)\).

This characteristic of a function is as fundamental as breathing. It ensures we always know where we can chart a plot and understand the boundaries or breadth of potential inputs.