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Chapter 3: Problem 72

Find, without graphing, where each of the given functions is continuous. $$ \frac{x-1}{x^{2}-1} $$

Short Answer

Expert verified
The function is continuous for all \( x \) except \( x = -1 \) and \( x = 1 \).

Step by step solution

01

Identify the Domain

The function given is \( f(x) = \frac{x-1}{x^2-1} \). This is a rational function, and it is defined everywhere except where the denominator is zero. So, first, set the denominator equal to zero and solve: \( x^2 - 1 = 0 \).
02

Solve for Zero Points in the Denominator

Factor the equation from Step 1: \( x^2 - 1 = (x+1)(x-1) = 0 \). Solve for \( x \) to find where the function is undefined: \( x = -1 \) and \( x = 1 \).
03

Identify the Discontinuities

The function is not defined at \( x = -1 \) and \( x = 1 \). These are the points of discontinuity for the function.
04

Determine Where the Function is Continuous

Since the function is not defined and thus discontinuous at the points where the denominator is zero, the function is continuous everywhere except at \( x = -1 \) and \( x = 1 \).

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Key Concepts

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

Rational Functions
A rational function is a quotient of two polynomials. For example, in the function \( f(x) = \frac{x-1}{x^2-1} \), the numerator is a polynomial \( x-1 \) and the denominator is another polynomial \( x^2-1 \). Rational functions are quite common and they can describe many real-world situations where ratios or proportional relationships are involved. Understanding these functions is key in calculus because they behave predictably in many ways, except at certain critical points.

When dealing with a rational function, it's important to remember:
  • They are defined everywhere except where their denominators are zero. This gives rise to potential discontinuities, often referred to as vertical asymptotes.
  • They can often be simplified to look more manageable, but care must be taken to understand the implications on the domain.
Discontinuities
Discontinuities are points where a function is not smooth or where it does not have a defined value. In rational functions, discontinuities occur where the denominator equals zero, leading the function to be undefined. For the function \( f(x) = \frac{x-1}{x^2-1} \), finding these discontinuities requires solving for when the denominator is zero.

To identify discontinuities:
  • Set the denominator to zero and solve for \( x \). For this function, the equation \( x^2 - 1 = 0 \) is solved by factoring to \( (x+1)(x-1) = 0 \).
  • Solutions \( x = -1 \) and \( x = 1 \) indicate where the function is undefined, and these are your points of discontinuity.
Remember that at these points, the function can exhibit different behaviors, such as jumping between values or reaching infinity.
Domain of a Function
The domain of a function refers to all the input values (or "x-values") for which the function is defined. For rational functions, the domain includes all real numbers except those that cause the denominator to be zero.

Understanding the domain of \( f(x) = \frac{x-1}{x^2-1} \):
  • The function is defined for all real numbers except where \( x^2-1 = 0 \). These are the values we calculated as \( x = -1 \) and \( x = 1 \).
  • The domain can be expressed as all real numbers \( x \) where \( x eq -1 \) and \( x eq 1 \).
Knowing the domain helps avoid any undefined points and ensures that analysis such as limits or derivatives are handled correctly. It provides a clear picture of where the function operates smoothly.

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Most popular questions from this chapter

If the side of a cube decreases from 10 feet to 9.6 feet, use the linear approximation formula to estimate the change in volume. Use a computer or graphing calculator to estimate any slope that you need.

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Tax Revenue Suppose that \(R(x)\) gives the revenue in billions of dollars for a certain state when the income tax is set at \(x \%\) of taxable income. Suppose that when \(x=3,\) the instantaneous rate of change of \(R\) with respect to \(x\) is 2 . Explain what this means.

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