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Chapter 1: Problem 88

Find the domain of the given function. Express the domain in interval notation. $$p(x)=(x-1)^{2}\left(x^{2}-9\right)^{-35}$$

Short Answer

Expert verified
The domain is \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

Step by step solution

01

Identify the Function Components

The given function is a product of two terms: \((x-1)^{2}\) and \((x^{2}-9)^{-35}\). The first term, \((x-1)^2\), is a polynomial and is defined for all real numbers. The second term, \((x^{2} - 9)^{-35}\), is a rational expression due to the negative exponent, which implies a division. This term is undefined where its base, \(x^{2} - 9\), equals zero.
02

Find Undefined Points for Second Term

Set the base of the exponent in the second term to zero and solve for \(x\): \[ x^2 - 9 = 0 \] Add 9 to both sides: \[ x^2 = 9 \] Take the square root of both sides: \[ x = 3 \text{ or } x = -3 \]. These are the points where the function is undefined because division by zero would occur.
03

Determine the Domain Excluding Undefined Points

Since the function is undefined at \(x = 3\) and \(x = -3\), these points must be excluded from the domain. Therefore, the domain is all real numbers except where \(x = 3\) and \(x = -3\). In interval notation, this can be expressed as: \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Understanding how to express a set of numbers in interval notation is crucial in defining the domain of a function. Interval notation provides a concise way to describe a range of values by using parentheses \(()\) and brackets \([]\) to indicate whether endpoints are included or not.
  • Parentheses \(()\) denote that an endpoint is not included, known as an "open interval." For example, the interval \((3, 5)\) includes all numbers greater than 3 and less than 5, but not 3 or 5 themselves.
  • Brackets \([]\) indicate that an endpoint is included, which is a "closed interval." Thus, \([3, 5]\) includes 3, 5, and all numbers in between.
  • In the given problem, the domain excludes two specific points, \(x = 3\) and \(x = -3\). Therefore, these points are expressed in interval notation using open intervals to exclude \(3\) and \(-3\): \((-infty, -3) \cup (-3, 3) \cup (3, infty)\).
This standardized way of expressing intervals helps students and mathematicians alike to communicate clearly about the range of values a function can take.
Rational Expressions
Rational expressions are fractions where both the numerator and denominator are polynomials. In the function given, \(\((x^{2} - 9)^{-35}\)\), the negative exponent implies a division, transforming it into a rational expression. This means we are effectively dealing with \((x^{2} - 9)^{-35}\) as \( rac{1}{(x^{2} - 9)^{35}}\).
  • A rational expression is undefined wherever its denominator equals zero. This occurs because division by zero is mathematically undefined and leads to infinite or indeterminate results.
  • To ensure the function is properly defined, it's essential to identify and exclude these points where the denominator is zero, which in this case were found to be \(x = 3\) and \(x = -3\).
By recognizing where rational expressions become undefined, you can accurately determine the exclusive points from the domain of a function.
Undefined Points
Undefined points in a function are points where the function does not exist. These often occur in rational expressions where division by zero can happen.
  • To find the undefined points, solve the equation where the denominator equals zero. For example, in \(\((x^{2} - 9)^{-35}\)\), setting \(x^{2} - 9 = 0\) provides the undefined points.
  • Solving the equation \(\x^2 = 9\)\ gives the solutions \(x = 3\)\ and \(x = -3\)\.
  • These values create a condition where the original function cannot be solved, hence it is undefined at these points.
Recognizing and writing around undefined points is a critical step in determining the applicable domain of a function. Removing these points from the domain ensures the function remains valid and usable over its defined range.

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