Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$f(x)=-3$$

Step-by-Step Solution

Answer

The line is horizontal at \(y = -3\), no x-intercept, y-intercept at (0, -3), domain: all real numbers, range: \(-3\), slope: 0.

Step 1: Identify the Type of Line

The given function is \(f(x) = -3\), which represents a horizontal line on a graph because it is a constant function.

Step 2: Plotting the Horizontal Line

Since the function is \(f(x) = -3\), plot a horizontal line that goes through all points where \(y = -3\) on the coordinate graph.

Step 3: Finding the x-Intercept

The line \(y = -3\) does not cross the x-axis; hence, there is no x-intercept.

Step 4: Finding the y-Intercept

The y-intercept is the point where the line crosses the y-axis. Since the line is \(y = -3\), it crosses the y-axis at the point \((0, -3)\).

Step 5: Determine the Domain

The domain of \(f(x) = -3\) includes all real numbers since \(x\) can take any value.

Step 6: Determine the Range

The range is the single value \(-3\) because the function represents a horizontal line at \(-3\).

Step 7: Calculate the Slope

The slope of a horizontal line is always 0 because there is no change in the \(y\)-values.

Key Concepts

Understanding the x-intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. This is a valuable point because at this position, the y-coordinate is zero. However, for horizontal lines like the one given by the function \( f(x) = -3 \), which runs parallel to the x-axis, there is no crossing, meaning it does not touch the x-axis at any point. As a result, horizontal lines do not have an x-intercept. Always remember, if a line is entirely horizontal and doesn’t touch the x-axis, it lacks an x-intercept.

Understanding the y-intercept

The y-intercept of any line is the point where the line crosses the y-axis. For the function \( f(x) = -3 \), which represents a horizontal line, the y-intercept is straightforward to identify. Since all points on this line have a y-value of -3, the y-intercept is found where \( x = 0 \). Therefore, the y-intercept of this line is the point

• \( (0, -3) \).

Whenever you encounter a horizontal line, simply look for the constant value in the function’s equation; that will be your y-intercept.

Domain and Range of Horizontal Lines

Horizontal lines can be interesting in terms of their domain and range. Let's start with the domain. When we talk about the domain of a function, we refer to all the possible x-values that can be plugged into the function. For the horizontal line \( f(x) = -3 \), there are no restrictions on the x-values. This means the domain includes all real numbers, so any x-value is valid.

When it comes to the range of horizontal lines, things are slightly different. The range signifies all the possible y-values the function can attain. Since this line remains constant at \( y = -3 \) for every point, the range is just

• \( \{-3\} \).

In essence, horizontal lines have a vast domain but a restricted range.

Examining the Slope of a Line

The slope of a line is a measure of its steepness and is calculated as the change in y divided by the change in x (\(\frac{\Delta y}{\Delta x}\)). For horizontal lines, like the one in our function \( f(x) = -3 \), the concept of slope is very interesting. Since the line stays at a constant y-value, there is no change in y as you move along the line, giving you a change of zero. This results in a slope of

• 0
Anytime you come across a horizontal line, remember that its slope will always be zero. It's a handy rule to keep in your mathematical toolbox; horizontal lines are flat and have no incline.