Problem 46 - College Algebra Essentials

Question

the domain of each piecewise function is \((-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$ f(x)=\left\\{\begin{array}{lll} \frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The plot of the function consists of a line with slope 0.5 on the left of and including the y-axis, and a horizontal line at \(y = 3\) on the right of the y-axis. Its range is \((-\infty, 3] \cup [3, +\infty)\).

Step 1: Plot the first piece

The first piece of the function, \(f(x) = 0.5x\), applies when \(x \leq 0\). This is a straight-line function, meaning it can be graphed as a line. The slope of the line is 0.5. Since this piece applies for \(x \leq 0\), the line should be drawn on the left of and including the y-axis.

Step 2: Plot the second piece

The second piece of the function, \(f(x) = 3\), applies when \(x > 0\). This is a constant function and is graphed as a horizontal line at y = 3. Since this piece applies for \(x > 0\), the line is drawn on the right of the y-axis.

Step 3: Determine the range

The range of a function is the set of output values (y-values). By looking at the graph, we can see that the function covers all y-values from \(-\infty\) up to 3, and then exactly 3 for all \(x > 0\).

Key Concepts

Function Graphing

When it comes to graphing piecewise functions, it might seem a bit tricky at first, but once you understand the process, it becomes much simpler. A piecewise function is made up of multiple "pieces," each defined by different expressions depending on the input value, or the domain of that piece. To graph a piecewise function accurately:

• Sketch each "piece" of the function separately. Start by identifying the domain of each part — that is, the range of x-values for which the piece is defined.
• For linear pieces, such as the part of the function described by \(f(x) = 0.5x\), draw a straight line. Use the slope-intercept form to help you plot this easily, where the formula is \(y = mx + b\), with \(m\) as the slope.
• For constant pieces, as seen with \(f(x) = 3\), draw a horizontal line across the specified x-values.

Remember that each piece of the function graph is distinct, adhering strictly to its domain, and meeting at exactly one point on the graph.

Domain and Range

The domain and range are key concepts when analyzing functions.

• The domain of a function is the set of all possible input values (x-values). For piecewise functions like the one given, the domain can often span all real numbers because each piece applies to specific intervals. In this case, the domain is \((-\infty, \infty)\).
• The range of a function is the set of all possible output values (y-values). In the exercise, you can determine the range by observing the sections of the graph. For the piecewise function \(f(x)\), the range from the graph goes from horizontal levels coinciding with the y-values of each piece.

By examining the graph, you will see the y-values extend from very negative to \(-\infty\) and then shift to exactly 3 for all x-values greater than zero. This gives the function a range of \((-\infty, 3]\).

Linear Functions

Linear functions are the backbone of analyzing the piecewise function given. They are characterized by a constant rate of change, making them easy to predict and graph.

• In the case of \(f(x) = 0.5x\), this linear function defines how the output (y-value) changes with the input (x-value) for \(x \leq 0\).
• The slope tells us how steep the line is. A slope of 0.5 means for every increase of 1 unit in x, the value of y will increase by 0.5.
• Linear functions can often be described by the slope-intercept form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept — the value of y when \(x = 0\).

Understanding the properties of linear functions helps when plotting them as parts of a piecewise function graph, ensuring accuracy and clarity in your mathematical work.