Problem 11 - Algebra and Trigonometry

Question

\(9-18\) m function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .(b)\) Find the domain and range of \(f\) from the graph. $$ f(x)=4, \quad 1 \leq x \leq 3 $$

Step-by-Step Solution

Answer

Domain: \([1, 3]\); Range: \{4\}.

Step 1: Understanding the Function

The function given is constant: \( f(x) = 4 \). This means that for every value of \( x \) in the domain of the function, the output will always be 4.

Step 2: Graph the Function

To graph the function \( f(x) = 4 \) for \( 1 \leq x \leq 3 \), draw a horizontal line at \( y = 4 \) from \( x = 1 \) to \( x = 3 \) on a graphing calculator or graphing software.

Step 3: Identify the Domain

The domain of the function \( f(x) = 4 \) as given is \( 1 \leq x \leq 3 \). Therefore, the domain is the closed interval \([1, 3]\). This means \( x \) can take any value between 1 and 3, inclusive.

Step 4: Identify the Range

The output of the function \( f(x) = 4 \) is always 4, regardless of the value of \( x \) in the domain. Therefore, the range of the function is simply \{4\}.

Key Concepts

Domain

The concept of a domain in mathematics refers to all the possible input values (usually denoted as "x") that a function can accept. These input values define where the function exists on the x-axis. For the function given in the exercise, which is a constant function, the domain is specified as the interval \(1 \leq x \leq 3\). This means that this function can accept any "x" value that ranges from 1 to 3, inclusive. We often denote this as a closed interval [1, 3] because the values 1 and 3 are included in the set of possible inputs.

Understanding the domain of a function is crucial because it tells us the span of "x" on the graph. In this case, any value outside of 1 to 3 is not considered part of the function. If you try to enter an x value like 0 or 4 into this function, it would not meet the specified domain. Hence, when graphing or using a function, it's important to pay close attention to these bounds.

If you're working manually or using graphing software, always ensure that you set your x-axis to appropriately reflect the domain. In practice, especially with functions that aren't constant, understanding the domain helps to avoid errors with undefined or non-real results.

Range

The range of a function defines all the possible output values, which are typically represented on the y-axis of a graph. For the constant function given in this exercise, \(f(x) = 4\), the range is quite simple. No matter what x-value you substitute within the domain, the output (or y-value) remains 4.

This characteristic makes constant functions easy to comprehend, as their range consists of just a single value. Here, the range is represented by the set \{4\}. This set indicates that the only output is 4, reinforcing that the graph is a horizontal line on y=4 across the entire domain [1, 3].

In typical functions with varying x-values leading to different outputs, the range can be a bit more complex. However, for a constant function such as this, it showcases the simple beauty of a single, unchanging output value. Recognizing this pattern in constant functions can save time and effort when identifying the range, directly translating into a more efficient analysis of other mathematical concepts.

Graphing Calculator

Graphing calculators are powerful tools that allow us to visualize mathematical functions quickly and accurately. In this exercise, using a graphing calculator to graph the constant function \(f(x) = 4\) involves plotting a horizontal line at y=4 between x=1 and x=3.

• First, input the function into the calculator. Many calculators will have a specific function button or menu for entering y=mx+b equations. For constant functions, you would enter y=4.
• Set the window to display the x-range from 1 to 3, as this aligns with the domain of the function.
• Finally, ensure that the y-axis reflects the constant output value for clear visualization, like setting the window's y-min to values below and y-max to values above 4.

With the line plotted, the graphing calculator visually represents the function's domain and range. It helps confirm our understanding that no matter the value of x, within the specified domain, the output remains constant. Utilizing a graphing calculator not only confirms solution accuracy but also develops a deeper understanding of function behavior in an intuitive manner.