Problem 26 - 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). $$f(x)=1.25$$

Step-by-Step Solution

Answer

Horizontal line; No \(x\)-intercept, \(y\)-intercept (0, 1.25); Domain \(\mathbb{R}\); Range \{1.25\}; Slope 0.

Step 1: Understand the Function

The function given is \(f(x) = 1.25\), which means that no matter what \(x\)-value is, the \(y\)-value is always 1.25. This represents a horizontal line.

Step 2: Determine the x-intercept

The \(x\)-intercept is the point where the function crosses the \(x\)-axis, i.e., where \(f(x) = 0\). For the function \(f(x) = 1.25\), since \(y\) is always 1.25 and never 0, there is no \(x\)-intercept.

Step 3: Determine the y-intercept

The \(y\)-intercept is the point where the function crosses the \(y\)-axis, when \(x = 0\). Here, \(f(0) = 1.25\), so the \(y\)-intercept is at the point (0, 1.25).

Step 4: Determine the Domain

The domain of a function is the set of all possible \(x\)-values. Since \(f(x) = 1.25\) is a horizontal line that exists for all \(x\), the domain is all real numbers, \(\mathbb{R}\).

Step 5: Determine the Range

The range of a function is the set of all possible \(y\)-values. Here, \(f(x) = 1.25\) means \(y\) is always 1.25, so the range is just \{1.25\}.

Step 6: Calculate the Slope

The slope of a line is the change in \(y\) over the change in \(x\). For a horizontal line like \(f(x) = 1.25\), there is no change in \(y\), so the slope is 0.

Key Concepts

Understanding the X-intercept

In mathematics, the x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept, we determine the point at which the y-value of a function is zero, which occurs at the x-axis. However, not all lines have an x-intercept. A horizontal line, such as the function given by \(f(x) = 1.25\), doesn't intersect the x-axis because its y-value is always above or below the axis. Hence, \(f(x) = 1.25\) has no x-intercept.

When analyzing any function, it's essential to check if these unique points exist as they can influence the overall graph and interpretation of the line.

Exploring the Y-intercept

The y-intercept of a line is where the line crosses the y-axis. For any function, this is the point where \(x = 0\). To find the y-intercept, you substitute \(x = 0\) into the function. For our given function \(f(x) = 1.25\), substituting \(x = 0\) results in \(f(0) = 1.25\).

Thus, the y-intercept of the function is the point \((0,1.25)\). This means the line crosses the y-axis at 1.25 on the y-scale. Y-intercepts offer valuable insight into the initial value of a function when \(x\) is zero and are used frequently in understanding linear equations.

• The y-intercept represents a starting value in many real-life scenarios.
• It's useful for quick graph plotting.
• In equations, this is usually the constant term.

Diving into Domain and Range

The domain and range describe the possible values for \(x\) and \(y\) respectively. These concepts help frame the boundaries within which a function operates. Let's break them down:

• **Domain:** The domain of a function is the collection of all possible values that \(x\) can take. For a horizontal line like \(f(x) = 1.25\), the domain is all real numbers, \(\mathbb{R}\). This means you can plug any real number into the function and consistently get a valid output.
• **Range:** The range is the set of all possible output values (or \(y\) values) of a function. For \(f(x) = 1.25\), the range is simply \(\{1.25\}\) since the output is always 1.25 regardless of the input.

Understanding these sets helps in visualizing and graphing lines properly, ensuring that the graph encapsulates all realistic scenarios the line could represent.

Deciphering the Slope of a Line

The slope of a line measures the steepness and direction of the line. It is calculated as the ratio of the vertical change to the horizontal change between two points on a line. Mathematically, this is often expressed as \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta\) symbolizes a difference or change in values.

For the function \(f(x) = 1.25\), which represents a horizontal line, the y-value does not change as x varies. Therefore, \(\Delta y = 0\), resulting in a slope (\(m\)) of 0. In simpler terms, horizontal lines like \(f(x) = 1.25\) have no tilt, hence a slope of zero.

A zero slope signifies:

• The line is perfectly flat.
• No rise or decline over its course.
• Functions with a zero slope align parallel to the x-axis.

Recognizing the slope is crucial for graphing lines and understanding their behavior and relationship to changes in input values.