Which one of the following is correct in respect of the graph of \(\rm y = \dfrac{1}{x-1} ?\)
The domain is {x ∈ R| x ≠ 1} and the range is the set of points on the y-axis except y = 0
Analyzing the Graph of \( \rm y = \dfrac{1}{x-1} \)
The question asks about the properties of the graph of the function \( \rm y = \dfrac{1}{x-1} \), specifically its domain and range. This is a rational function.
Determining the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the function is undefined when the denominator is equal to zero.
In the function \( \rm y = \dfrac{1}{x-1} \), the denominator is \( \rm (x-1) \). To find the values of \( \rm x \) for which the function is undefined, we set the denominator to zero:
\( \rm x - 1 = 0 \)
Solving for \( \rm x \):
\( \rm x = 1 \)
Therefore, the function is undefined when \( \rm x = 1 \). The domain of the function is all real numbers except \( \rm 1 \).
In set notation, the domain is \( \rm \{x \in \mathbb{R} | x \neq 1\} \).
Determining the Range
The range of a function is the set of all possible output values (y-values) that the function can take. To find the range of \( \rm y = \dfrac{1}{x-1} \), we can consider the equation \( \rm y = \dfrac{1}{x-1} \) and try to express \( \rm x \) in terms of \( \rm y \).
Multiply both sides by \( \rm (x-1) \) (assuming \( \rm x \neq 1 \)):
\( \rm y(x-1) = 1 \)
Distribute \( \rm y \):
\( \rm yx - y = 1 \)
Add \( \rm y \) to both sides:
\( \rm yx = 1 + y \)
Now, solve for \( \rm x \) by dividing by \( \rm y \). However, we can only divide by \( \rm y \) if \( \rm y \neq 0 \).
- If \( \rm y \neq 0 \), we get \( \rm x = \dfrac{1+y}{y} \). For every real value of \( \rm y \) except 0, there is a corresponding real value of \( \rm x \).
- If \( \rm y = 0 \), the original equation becomes \( \rm 0 = \dfrac{1}{x-1} \). This equation has no solution because \( \rm 1 \) can never be equal to \( \rm 0 \). This means that \( \rm y \) can never be equal to \( \rm 0 \).
Therefore, the range of the function is all real numbers except \( \rm 0 \).
In set notation, the range is \( \rm \{y \in \mathbb{R} | y \neq 0\} \). This corresponds to the set of points on the y-axis except \( \rm y = 0 \).
Analyzing the Options
Let's examine each option based on our findings for the domain and range of \( \rm y = \dfrac{1}{x-1} \).
- Option 1: "The domain is {x ∈ R| x ≠ 1} and the range is the set of reals."
The domain part is correct. However, the range is stated as the set of all real numbers, which is incorrect because \( \rm y \) cannot be \( \rm 0 \).
- Option 2: "The domain is {x ∈ R| x ≠ 1}, the range is {y ∈ R| y ∈ 0} and the graph intersects y-axis at (0, -1)."
The domain part is correct. The range notation {y ∈ R| y ∈ 0} is incorrect and misleading (it suggests y must be 0). While the y-intercept at (0, -1) is correct (set \( \rm x=0 \), \( \rm y = \dfrac{1}{0-1} = -1 \)), the range statement makes this option incorrect.
- Option 3: "The domain is the set of reals and the range is the singleton set {0}."
The domain is stated as the set of all real numbers, which is incorrect because \( \rm x \neq 1 \). The range is stated as only the value 0, which is also incorrect.
- Option 4: "The domain is {x ∈ R| x ≠ 1} and the range is the set of points on the y-axis except y = 0"
The domain is correctly stated as \( \rm \{x \in \mathbb{R} | x \neq 1\} \). The range is correctly stated as the set of points on the y-axis except \( \rm y = 0 \), which means \( \rm \{y \in \mathbb{R} | y \neq 0\} \).
Based on our analysis, Option 4 correctly describes both the domain and the range of the function \( \rm y = \dfrac{1}{x-1} \).
| Property | Analysis for \( \rm y = \dfrac{1}{x-1} \) | Result |
|---|---|---|
| Domain | Denominator \( \rm (x-1) \) cannot be 0. Thus, \( \rm x \neq 1 \). | \( \rm \{x \in \mathbb{R} | x \neq 1\} \) |
| Range | Solving for \( \rm x \) in \( \rm y = \dfrac{1}{x-1} \) leads to \( \rm x = \dfrac{1+y}{y} \). This is defined for all \( \rm y \) except \( \rm y = 0 \). Also, \( \rm y=0 \) is not possible from the original equation. | \( \rm \{y \in \mathbb{R} | y \neq 0\} \) |
Conclusion
The correct description for the graph of \( \rm y = \dfrac{1}{x-1} \) is that its domain is all real numbers except 1, and its range is all real numbers except 0.
Revision Table: Key Concepts for Graphing \( \rm y = \dfrac{1}{x-1} \)
| Concept | Explanation | Application to \( \rm y = \dfrac{1}{x-1} \) |
|---|---|---|
| Rational Function | A function that can be written as the ratio of two polynomials, \( \rm f(x) = \dfrac{P(x)}{Q(x)} \), where \( \rm Q(x) \neq 0 \). | \( \rm y = \dfrac{1}{x-1} \) is a rational function with \( \rm P(x) = 1 \) and \( \rm Q(x) = x-1 \). |
| Domain | The set of all valid input values (\( \rm x \)). For rational functions, exclude values where the denominator is zero. | Exclude \( \rm x \) where \( \rm x-1=0 \), so \( \rm x \neq 1 \). Domain: \( \rm \{x \in \mathbb{R} | x \neq 1\} \). |
| Vertical Asymptote | A vertical line \( \rm x=a \) where the graph approaches infinity. Occurs at values of \( \rm x \) that make the denominator zero but not the numerator (after cancelling common factors). | Denominator is zero at \( \rm x=1 \), numerator is not zero. Vertical asymptote at \( \rm x=1 \). |
| Horizontal Asymptote | A horizontal line \( \rm y=b \) that the graph approaches as \( \rm x \) approaches \( \pm \infty \). Determined by the degrees of the numerator and denominator polynomials. | Degree of numerator (0) is less than the degree of the denominator (1). Horizontal asymptote at \( \rm y = 0/1 = 0 \). |
| Range | The set of all possible output values (\( \rm y \)). Often related to horizontal asymptotes and local extrema. | Graph approaches \( \rm y=0 \) but never reaches it. Range: \( \rm \{y \in \mathbb{R} | y \neq 0\} \). |
Additional Information: Graphing Rational Functions
Understanding domain and range is crucial for sketching the graph of a rational function like \( \rm y = \dfrac{1}{x-1} \).
- The point \( \rm x=1 \) is a vertical asymptote. This means the graph gets infinitely close to the vertical line \( \rm x=1 \) but never touches or crosses it.
- The line \( \rm y=0 \) (the x-axis) is a horizontal asymptote. This means as \( \rm x \) goes to positive or negative infinity, the graph gets infinitely close to the x-axis but never touches or crosses it (in this specific case, it doesn't cross).
- The domain restriction \( \rm x \neq 1 \) corresponds directly to the location of the vertical asymptote.
- The range restriction \( \rm y \neq 0 \) corresponds directly to the location of the horizontal asymptote.
- The graph of \( \rm y = \dfrac{1}{x-1} \) is a hyperbola, shifted 1 unit to the right compared to the basic graph of \( \rm y = \dfrac{1}{x} \). The graph has two branches, one in the top-right region formed by the asymptotes, and one in the bottom-left region.
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