/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 127 My graph of \(y=\frac{x-1}{(x-1)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 127

My graph of \(y=\frac{x-1}{(x-1)(x-2)}\) has vertical asymptotes at \(x=1\) and \(x=2\).

Short Answer

Expert verified
The function \(y=\frac{x-1}{(x-1)(x-2)}\) does indeed have vertical asymptotes at x=1 and x=2.

Step by step solution

01

Set the denominator equal to zero

To find out where the function could be undefined, you solve \( (x-1)(x-2)=0 \). The solutions to this equation will give us the x values for the vertical asymptotes.
02

Solve for x

By solving the equation, you would find that x can be 1 or 2. This essentially means that x=1 and x=2 are the vertical asymptotes.
03

Confirm the answer

Take a look at the graph for the function \(y=\frac{x-1}{(x-1)(x-2)}\). You will notice vertical asymptotes at x=1 and x=2, confirming the solution is correct.

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Key Concepts

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

Understanding Undefined Functions
A function is undefined when its expression leads to an operation that is mathematically impossible. In most cases, this involves division by zero. For rational functions, which are the ratio of two polynomials, a problem arises whenever the denominator equals zero.
To identify this, it's crucial to find which values of the variable make the denominator zero, as this causes the function to be undefined. For example, in our given function, which is \( y = \frac{x-1}{(x-1)(x-2)} \), the denominator is \((x-1)(x-2)\).
When the denominator is zero, none of the quantities involved have defined numerical values, and we can no longer calculate the output of the function.
  • Substitute values into the denominator and check where it leads to zero.
  • Those values indicate where the function becomes undefined.
  • This helps in identifying potential vertical asymptotes.
Denominator Zero Condition
The denominator zero condition helps determine where a rational function encounters vertical asymptotes. This occurs specifically when the denominator equals zero, but the numerator does not simultaneously become zero at the same points.
To apply the concept, solve for the values of \(x\) that make the denominator zero. In our function, \((x-1)(x-2)=0\), revealing that \(x=1\) and \(x=2\) both result in a zero denominator.
  • Set the denominator equal to zero.
  • Solve the resulting equation to find the problematic \(x\) values.
  • In our example, these values are 1 and 2.
This highlights where the function breaks down in terms of continuity and is a vital step in exploring its behavior. However, we must ensure that the numerator is not also zero at these points, to definitively identify vertical asymptotes.
Graphing Rational Functions with Vertical Asymptotes
Graphing rational functions like \(y = \frac{x-1}{(x-1)(x-2)}\) involves clearly marking where vertical asymptotes occur. Vertical asymptotes are shown as dashed vertical lines on the graph, indicating where the output of the function skyrockets towards infinity or plunges towards negative infinity.
As a rule, vertical asymptotes occur at the \(x\) values where the function is undefined (due to a zero denominator) yet the numerator does not cancel out entirely.
  • Identify any points where the vertical asymptotes lie, such as \(x=1\) and \(x=2\) in our example.
  • Use graphs to illustrate how these points affect the behavior of the function.
  • Notice that around each vertical asymptote, the function approaches a very steep curve.
This process helps elucidate how the function behaves near undefined values and provides a visual aid that can enhance understanding of complex rational functions.

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Most popular questions from this chapter

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-1}$$

You drive from your home to a vacation resort 600 miles away. You return on the same highway. The average velocity on the return trip is 10 miles per hour slower than the average velocity on the outgoing trip. Express the total time required to complete the round trip, \(T\), as a function of the average velocity on the outgoing trip, \(x .\)
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