Exploring the Domain and Range of the Function h(x) 4 - x

In the realm of mathematics, particularly in algebra and calculus, the domain and range of a function are fundamental concepts. This article delves into the domain and range of the specific function h(x) 4 - x, exploring its behavior and visual representation.

Introduction to the Function

The function h(x) 4 - x is a simple linear function that can be easily graphed and analyzed. This type of function, known as a linear function, can be expressed in the form y mx b, where m is the slope and b is the y-intercept. In the context of h(x) 4 - x, the slope m is -1 and the y-intercept b is 4.

Understanding the Domain and Range

Domain of h(x) 4 - x

The domain of a function refers to the set of all possible input values (independent variables) for which the function is defined. For the function h(x) 4 - x, the domain encompasses all real numbers. This is because the function can accept any real number as input and produce a corresponding output. In mathematical notation, this is represented as:

Domain (D): D (-∞, ∞)

Range of h(x) 4 - x

The range of a function, on the other hand, is the set of all possible output values (dependent variables) that the function can produce. For the function h(x) 4 - x, the range also extends to all real numbers, as the function can yield any real number as an output. The range is mathematically represented as:

Range (R): R (-∞, ∞)

Calculating Specific Values for h(x) 4 - x

To further illustrate the behavior of the function, let's calculate the values for specific inputs. The given examples show the function for integer values from 0 to 5:

h(0) 4 - 0 4

When the input is 0, the output is 4. This corresponds to the y-intercept of the function's graph, confirming the y-intercept as 4.

h(1) 4 - 1 3

When the input is 1, the output is 3.

h(2) 4 - 2 2

When the input is 2, the output is 2.

h(3) 4 - 3 1

When the input is 3, the output is 1.

h(4) 4 - 4 0

When the input is 4, the output is 0.

h(5) 4 - 5 -1

When the input is 5, the output is -1.

Summary of Calculations

We can encapsulate these specific calculations as follows:

h(0) 4 h(1) 3 h(2) 2 h(3) 1 h(4) 0 h(5) -1

Graph of h(x) 4 - x

A visual representation of the function h(x) 4 - x can be very helpful for understanding its behavior. The graph of this function is a straight line with a slope of -1 and a y-intercept of 4. The points calculated above can be plotted on the graph, as shown:

(0, 4) (1, 3) (2, 2) (3, 1) (4, 0) (5, -1)

These points, when connected, form a line that extends infinitely in both directions, confirming the unbounded nature of both the domain and range.

Conclusion

The function h(x) 4 - x is a simple yet crucial concept in algebra. Its domain and range both extend to all real numbers, reflecting its linear nature. Understanding the domain and range is fundamental for graphing and analyzing functions in mathematics. Whether you are a student, a teacher, or a professional, grasping these concepts can provide a strong foundation in algebraic and analytical skills.

Related Resources

To further explore this topic and similar concepts, consider checking out:

Linear Functions - Math is Fun Intercepts Review - Khan Academy Algebra Learning Resources