Problem 47 - Algebra A Combined Function

Question

Draw a graph whose domain is \((-\infty, 5]\) and whose range is \([2, \infty)\). Is your graph a function? Discuss why or why not.

Step-by-Step Solution

Answer

Yes, it is a function because each x-value maps to one y-value.

Step 1: Understanding the Domain

The domain of the function is given as \((-\infty, 5]\),which means that our graph will include all x-values less than or equal to 5. This suggests that we'll start our graph from negative infinity and endpoint at some point on the x-axis at x = 5.

Step 2: Understanding the Range

The range is given as \([2, \infty)\),which indicates that the graph should only include y-values that are greater than or equal to 2. This means the graph should be starting from y = 2 and going upwards to infinity.

Step 3: Choosing the Type of Graph

Considering the domain and range, we need a graph that can infinitely cover negative x-values up to 5 and start at y = 2 to infinity. A suitable function could be a horizontal line at y = 2 extending to the left, up to x = 5. This suits the function's properties.

Step 4: Drawing the Graph

On a coordinate plane, begin by drawing a horizontal line from a point on the y-axis at y = 2, extending this line to the left until it reaches a point where x = 5. The line should stop exactly at x = 5 and should not continue to the right of this point. Ensure the line extends infinitely to the left, indicating negative infinity.

Step 5: Analyzing if the Graph is a Function

For the graph to be considered a function, every x-value in the domain should map to exactly one y-value. Because our graph is a horizontal line, every x-coordinate maps to the single value of y = 2, satisfying the definition of a function. Each x-value has exactly one y-value.

Key Concepts

Function Graph

When we talk about a function graph, we are essentially discussing a visual representation of a mathematical function. A function graph helps us see the relationship between variables, primarily between x-values (input) and y-values (output), on a two-dimensional coordinate plane.

• The domain represents all possible input values (x-values) which can be used in the function. It is depicted along the x-axis.
• The range represents all possible output values (y-values), displayed along the y-axis.

In this particular exercise, we are tasked with creating a graph that meets the domain (-∞, 5] and range [2, ∞). This means that in our visual illustration, the x-values begin at negative infinity and end at 5, while the y-values start at 2 and extend upwards without bound. The function graph in this kind of setup will help students understand the breadth and limitation of function values in a given mathematical model.

Horizontal Line

A horizontal line is a straight, level line that runs left to right. In a graph, it is characterized by all points having the same y-value regardless of the x-value. This concept is essential in many areas of math because horizontal lines represent constant functions.

• For a line to be horizontal, every point on the line has the same y coordinate.
• In our exercise, the horizontal line sits at y = 2, meaning y never changes regardless of x.

In our specific scenario, the graph stopping at x = 5 means that it includes all x-values less than or equal to 5, while y remains constant at 2. This type of graph satisfies the function's requirements since y = 2 is consistently maintained throughout the domain of x-values. It is also straightforward for students to understand since every x-value maps to the same y-value in a horizontal line.

Infinity in Domain and Range

Infinity in mathematics means a concept that represents an unbounded quantity that is larger than any real number. It is often used in discussing the domain and range in function graphs to explain unending or limitless extensions on the graph.

• An infinite domain indicates that the x-values extend indefinitely. For example, in the domain (-∞, 5], x-values start from negative infinity up to 5.
• An infinite range indicates y-values continue indefinitely. The range [2, ∞) means our function's y-value begins at 2 and extends to positive infinity.

Understanding how infinity operates in domain and range is vital for depicting and interpreting graphs correctly. It shows the extent and limitlessness of a function's reach, making it easier for students to grasp complex function behaviors on a graph.