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what is the range if y is undefined

what is the range if y is undefined

2 min read 21-01-2025
what is the range if y is undefined

What is the Range if Y is Undefined? Understanding Undefined Values in Functions

The question "What is the range if y is undefined?" hinges on understanding the concept of undefined values in mathematical functions, specifically how they affect the range (the set of all possible output values). Let's break this down.

Understanding Functions and Their Ranges

A function, in simple terms, is a relationship where each input (typically denoted by 'x') corresponds to exactly one output (typically denoted by 'y'). The set of all possible output values ('y' values) is called the range.

For example, consider the function f(x) = x². If we input various x-values, we get corresponding y-values:

  • x = 1, y = 1
  • x = 2, y = 4
  • x = -1, y = 1
  • x = 0, y = 0

The range of f(x) = x² is all non-negative real numbers (y ≥ 0) because squaring any real number always results in a non-negative number.

When 'y' is Undefined

'y' becomes undefined when the function's expression is invalid for certain input values ('x' values). This usually happens in these situations:

  • Division by zero: If the function involves division, and the denominator becomes zero for a particular 'x', then 'y' is undefined at that point. For example, in the function g(x) = 1/x, 'y' is undefined when x = 0.

  • Even roots of negative numbers: Taking the square root, fourth root, or any even root of a negative number is not defined within the realm of real numbers. For instance, in the function h(x) = √x, 'y' is undefined for x < 0.

  • Logarithms of non-positive numbers: The logarithm of a non-positive number is undefined. For example, in the function i(x) = log(x), 'y' is undefined for x ≤ 0.

Determining the Range When Undefined Values Exist

When 'y' is undefined for certain 'x' values, it means those 'y' values are excluded from the range. The range is still defined, but it's a subset of the possible outputs, excluding those where the function is undefined.

Example:

Let's consider the function g(x) = 1/x. We've already established that 'y' is undefined when x = 0. For all other x-values, 'y' can take on any value except zero. Therefore, the range of g(x) = 1/x is all real numbers except 0. We can write this as (-∞, 0) U (0, ∞) using interval notation.

Example 2 (Slightly More Complex):

Consider the function j(x) = √(x - 2). The expression under the square root must be non-negative, so x - 2 ≥ 0, which means x ≥ 2. When x < 2, 'y' is undefined. For x ≥ 2, the function produces non-negative values. Therefore, the range of j(x) is [0, ∞).

Conclusion

The range of a function when 'y' is undefined for some input values is the set of all possible output values excluding those where the function is invalid. To find the range, you need to:

  1. Identify points of discontinuity: Determine the values of 'x' where the function is undefined (division by zero, even roots of negative numbers, logarithms of non-positive numbers, etc.).

  2. Analyze the remaining values: Determine the range for the remaining 'x' values where the function is defined. Consider the function's behavior as 'x' approaches the points of discontinuity.

  3. Express the range: Write the range using interval notation or set builder notation, explicitly excluding the undefined 'y' values. Understanding the behavior of your function around undefined points is key to accurately determining the range.

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