what are the domain and range of the function below

Guri Bee

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20-10-2024

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When studying functions in mathematics, it's essential to understand the concepts of domain and range. The domain refers to all the possible input values (x-values) for a function, while the range consists of all the possible output values (y-values) that result from those inputs. In this article, we will explore how to find the domain and range of a function using a practical example. We'll also provide additional explanations, insights, and analysis to deepen your understanding.

Example Function

Let’s consider the function:

[ f(x) = \frac{1}{x - 2} ]

To find the domain and range of this function, we’ll follow a systematic approach.

Finding the Domain

1. Identify Restrictions: For the function ( f(x) = \frac{1}{x - 2} ), we must ensure that the denominator is not equal to zero since division by zero is undefined.

   [ x - 2 \neq 0 \quad \Rightarrow \quad x \neq 2 ]

2. State the Domain: Therefore, the domain of the function is all real numbers except for ( x = 2 ). In interval notation, we can express this as:

   [ \textDomain (-\infty, 2) \cup (2, \infty) ]

Finding the Range

1. Understand the Function Behavior: The function ( f(x) ) will approach positive or negative infinity as ( x ) approaches 2. For values of ( x ) very close to 2 (but not equal), ( f(x) ) will yield very large positive or negative values.

2. Determine Output Values: We can rewrite the function to observe its range:

   • As ( x ) approaches 2 from the left (( x \to 2^- )), ( f(x) \to -\infty ).
   • As ( x ) approaches 2 from the right (( x \to 2^+ )), ( f(x) \to +\infty ).
   • For all other x-values, ( f(x) ) can take on any real number except for zero, since there’s no x-value that will make ( f(x) = 0 ).

3. State the Range: Thus, the range of the function is all real numbers except for zero. In interval notation, we write:

   [ \textRange (-\infty, 0) \cup (0, \infty) ]

Summary

In summary, for the function ( f(x) = \frac{1}{x - 2} ):

• Domain: ( (-\infty, 2) \cup (2, \infty) )
• Range: ( (-\infty, 0) \cup (0, \infty) )

Additional Insights

Understanding the domain and range is crucial in graphing functions. For instance, when you know the domain, you can effectively narrow down the x-values you will plot. Similarly, knowing the range helps you predict the y-values that will appear in your graph.

Practical Examples

1. Real-Life Applications: In economics, the demand function often has a restricted domain, as negative quantities don't make sense. For example, if the demand function is ( D(p) = 100 - p ), where ( p ) is price, the domain is ( [0, 100] ) (as price cannot exceed 100, leading to negative demand).

2. Graphing: To visually validate the domain and range, sketch the graph of ( f(x) = \frac{1}{x - 2} ). You'll observe a vertical asymptote at ( x = 2 ) and a horizontal asymptote at ( y = 0 ), reinforcing our findings about the domain and range.

Conclusion

Understanding the domain and range of functions is an integral part of mathematical comprehension and is applicable in various fields, from science to economics. By practicing with different types of functions and their properties, you'll enhance your problem-solving skills and be better prepared for advanced mathematical studies.

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By utilizing clear examples and providing additional insights, this article aims to equip readers with a solid foundation in understanding the domain and range of functions, beyond the basic definitions. Happy learning!