Question

Multiple Choice Which of the following gives the range of \(f(x)=1+\frac{1}{x-1} ?\) \(\begin{array}{ll}{(\mathbf{A})(-\infty, 1) \cup(1, \infty)} & {(\mathbf{B}) x \neq 1} \\ {(\mathbf{D})(-\infty, 0) \cup(0, \infty)} & {(\mathbf{E}) x \neq 0}\end{array}\)

Short answer

The range of the function \(f(x)=1+\frac{1}{x-1}\) is all real numbers except for y = 1. None of the given choices accurately reflect this, so the question seems to be flawed. Such important observations about multiple choice exercises should be reported to the relevant authority.

Step-by-step solution

Analyze the Function

We first observe the function \(f(x) = 1 + \frac{1}{x-1}\). It is clear that the denominator of the fraction, \(x-1\), cannot equal zero, as this would result in division by zero, which is undefined. Thus, x cannot be 1. This is the only restriction on x.

Determine the Range

To determine the range, let's consider the end behavior of the function. As x approaches negative infinity or positive infinity, the fraction part of the function, \(\frac{1}{x-1}\), approaches 0, making the whole function approach 1. Next, let's observe what happens near x = 1. As x approaches 1 from the left, the function goes to negative infinity. Similarly, as x approaches 1 from the right, the function goes to positive infinity. Hence, the range of the function is all y-values except 1.

Key concepts

End Behavior of a Function

Understanding the end behavior of a function means determining how the function behaves as the input values, or 'x', approach infinity or negative infinity. This characteristic helps us predict the function's tendency in extreme cases, which is crucial for graphing and analyzing the function's long-term behavior.

For the given function, let's delve into the end behavior: \( f(x) = 1 + \frac{1}{x-1} \). As \( x \) approaches negative infinity (\( x \to -\infty \)), the fraction component \( \frac{1}{x-1} \) gets closer and closer to 0, but it's important to note that it remains negative. The '1' in the function doesn't change as \( x \) changes, so the overall function approaches 1. Similarly, as \( x \) tends to positive infinity (\( x \to \infty \)), the fraction part tends towards 0, and the value of the function again approaches 1. These observations are pivotal, as they inform us that the function doesn't extend beyond the value of 1 as \( x \) becomes very large or very small—in other words, the function's horizontal asymptote is \( y = 1 \) on both ends of the x-axis.

Asymptotes in Algebra

An asymptote is a line that a function approaches but never actually reaches, no matter how far the graph is extended. Asymptotes can be horizontal, vertical, or oblique (slanted), and they provide insight into the function's behavior at extreme values.

In the context of the given function, there is a vertical asymptote at \( x = 1 \). This is because the denominator of the fractional part, \( x-1 \), becomes zero when \( x = 1 \), and the function is not defined at this point. As mentioned in the analysis, as \( x \) gets very close to 1 from the left, the function's value dives toward negative infinity (\( -\infty \)), while it soars towards positive infinity (\( +\infty \)) when approached from the right.

Hence, vertical asymptotes serve as a boundary that the function values cannot cross and significantly affect the range, since the range will not include any values that correspond to the function shooting to infinity near the asymptotes.

Analyzing Rational Functions

Rational functions are ratios of polynomials and can present points of discontinuity where their denominators are zero, leading to undefined values. Analyzing a rational function like \( f(x) = 1 + \frac{1}{x-1} \) involves looking for these points of discontinuity, end behavior, and asymptotes, which affect the overall shape and range of the function's graph.

The behavior near the points of discontinuity and the end behavior inform the potential range of the function. By examining \( f(x) \) carefully, we know that the function is undefined at \( x = 1 \), which means that this value is not part of the domain. From there, we observed the end behavior and determined the horizontal asymptote at \( y = 1 \). Also, the lack of any values where \( f(x) \) equals exactly 1 indicates a hole in the range.

Therefore, for the given function, all y-values are possible except for 1, which gives us the range \( y eq 1 \). This consideration is crucial when identifying the correct multiple-choice answer that represents the range of such rational functions.