Determine Which Two Values The Following Expression Is Between.

Mohammad

5 min read

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Oct 07, 2024

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Determining the range of values an expression falls between is a fundamental skill in mathematics and programming. This process helps us understand the behavior of the expression, its potential values, and its impact within a larger context.

Understanding the Concept

The core idea is to find two specific values, one larger and one smaller, that enclose the result of the given expression. Imagine a number line; we aim to pinpoint a segment of the line where the expression's value lies.

Steps to Determine the Range

1. Simplify the expression: If the expression is complex, try simplifying it to make it easier to analyze.
2. Identify the key operations: Look for operations like addition, subtraction, multiplication, division, exponents, and functions. These operations influence the magnitude of the result.
3. Consider the potential values: Estimate the range of values each component of the expression could take on.
4. Analyze the impact of operations: How do the operations combine to affect the overall result?
5. Determine the upper and lower bounds: Based on your analysis, establish the two values that would enclose the expression's result.

Examples

Let's illustrate this with some examples:

Example 1:

• Expression: 2x + 5 where x is between 0 and 3.

• Analysis:

  • x ranges from 0 to 3.
  • Multiplying x by 2 would make the range from 0 to 6.
  • Adding 5 to the range shifts it to 5 to 11.

• Result: The expression 2x + 5 is between 5 and 11 when x is between 0 and 3.

Example 2:

• Expression: sqrt(x) where x is between 4 and 9.

• Analysis:

  • x ranges from 4 to 9.
  • Taking the square root of x would result in values between 2 and 3.

• Result: The expression sqrt(x) is between 2 and 3 when x is between 4 and 9.

Example 3:

• Expression: 1/x where x is between 1 and 2.

• Analysis:

  • x ranges from 1 to 2.
  • Dividing 1 by a value between 1 and 2 would result in values between 0.5 and 1.

• Result: The expression 1/x is between 0.5 and 1 when x is between 1 and 2.

Tips

• Visualize: It's often helpful to visualize the expression on a number line or graph.
• Consider extremes: Think about what happens to the expression when its components take on their minimum and maximum possible values.
• Use inequalities: You can formally express the range using inequalities, such as 5 < 2x + 5 < 11.

Conclusion

Determining the range of values an expression falls between is crucial for understanding its behavior and potential impact. By systematically analyzing the expression and the operations involved, we can confidently determine the upper and lower bounds of its possible values. This knowledge is essential for various applications in mathematics, computer science, and other disciplines.