Problem 29 - Algebra and Trigonometry

Question

A 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. $$ f(x)=4 $$

Step-by-Step Solution

Answer

Graph is a horizontal line at \( y = 4 \). Domain: \( (-\infty, \infty) \), Range: \( \{4\}. \)

Step 1: Understanding the Function

The function given is \( f(x) = 4 \). This is a constant function, which means that for every value of \( x \), \( f(x) \) remains the same, i.e., 4.

Step 2: Graphing the Function

Use a graphing calculator to draw the function \( f(x) = 4 \). This will be represented as a horizontal line on the graph at \( y = 4 \), passing through just above the x-axis, as all \( f(x) \) values are equal to 4.

Step 3: Finding the Domain from the Graph

The domain of a function refers to all possible input values (\( x \)-values) that the function can take. For \( f(x) = 4 \), \( x \) can be any real number since there are no restrictions indicated in the function or graph. Thus, the domain is \( (-\infty, \infty) \).

Step 4: Finding the Range from the Graph

The range of a function refers to all possible output values (\( y \)-values) that the function can produce. For the constant function \( f(x) = 4 \), the only output value is 4, regardless of the input. Hence, the range is \( \{4\} \).

Key Concepts

Understanding Domain and Range

The concepts of domain and range are essential in understanding functions. The domain of a function is the set of all possible input values, or the values you can substitute for the variable, usually denoted as \( x \). For most functions, especially those defined by basic algebraic equations, the domain includes all real numbers unless there are specific restrictions, such as division by zero or square roots of negative numbers. However, with a constant function like \( f(x) = 4 \), the domain is unrestricted.

For \( f(x) = 4 \), the domain is \((-\infty, \infty)\). This interval notation means there are no limits to the possible values of \( x \).

The range, meanwhile, is the set of all potential output values. For constant functions, the range is simple. Since the function's output does not change, it’s a single value. In this case, the range is just \( \{4\} \), indicating that no matter what \( x \) is, \( f(x) \) will always equal 4.

Graphing Constant Functions

Graphing a constant function is quite straightforward, yet it's a fundamental skill that helps in understanding more complex functions later. For the function \( f(x) = 4 \), the graph is a horizontal line parallel to the x-axis. This line will cross the y-axis at the point where \( y = 4 \).

When graphing \( f(x) = 4 \), the horizontal line means that no matter what point you choose along the x-axis, the corresponding y-value will always be 4. This visually represents the idea of a constant function—the output remains unchanged even as the input varies.

Using a graphing calculator can simplify this process. By entering the function \( f(x) = 4 \), the calculator generates the horizontal line automatically. It's crucial to interpret this graph correctly to understand both the domain and range without missing any detail.

Analyzing Constant Functions

Function analysis involves exploring the behavior and properties of mathematical functions. Constant functions, like \( f(x) = 4 \), offer a unique opportunity because they remain unchanged and are straightforward to analyze.

Some key points about constant functions include:

• They have a derivative of zero, since there is no change or slope in the function; the line is completely horizontal.


• The domain of these functions is always all real numbers \((-\infty, \infty)\) unless explicitly restricted.


• The range is a single value, highlighting their simplicity.

Understanding constant functions forms the basis for learning about more complicated functions. They teach us how to interpret graphs, evaluate domains and ranges, and appreciate the stability in mathematical relationships. Analyzing such a simple function builds a strong foundation to delve into the dynamics of other functions.