Understanding the Domain and Range of the Function y -x/5

When working with functions, it is crucial to understand the values that the variables can take, which brings us to the concepts of domain and range. In the case of the function y -x/5, let's explore how to determine its domain and range.

Domain of the Function

The domain of a function refers to the set of all possible input values (in this case, x) for which the function is defined. For the function y -x/5, we can substitute any real number into x. This is because the function is a simple division of x by 5, and there are no values of x that would make this division undefined. Therefore, the domain of this function is all real numbers.

Mathematical Representation of the Domain

The domain can be represented mathematically as:

Domain {x : x ∈ R}

This notation means that x can be any real number, and there are no restrictions on the values that x can take. The set of real numbers, denoted by R, includes all rational and irrational numbers.

Range of the Function

The range of a function is the set of all possible output values (in this case, y) that the function can produce. Given the function y -x/5, we can see that for any real number x, the output y will also be a real number. This is because the function is linear and continuous, and it spans all possible real numbers as x varies.

Mathematical Representation of the Range

The range of the function can be represented as:

Range {y : y ∈ R}

This means that y can also be any real number. The function y -x/5 will always produce a real number output, and there are no restrictions on the values that y can take. This is a characteristic of linear functions, which can span the entire set of real numbers.

Graphical Interpretation

A graphical representation of the function y -x/5 shows a straight line. The line has a slope of -1/5, which indicates that for every unit increase in x, y decreases by 1/5. The domain and range of a line can be determined by looking at the x and y-axis limits of the graph.

This graph demonstrates that the line extends indefinitely in both directions, confirming that the function's domain and range are all real numbers.

Conclusion

In summary, the domain and range of the function y -x/5 are both the set of all real numbers. This is a property of linear functions, where the domain and range can span all real values without any restrictions.

Additional Resources

For further exploration of the domain and range of functions, and a deeper understanding of linear functions, you may want to refer to resources such as online graphing tools or mathematical textbooks.