Problem 49
Question
Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=\frac{2}{x-1} $$
Step-by-Step Solution
Verified Answer
The function \(g(x) = \frac{2}{x-1}\) has a domain of \(\{x \in \mathbb{R} | x \neq 1\}\) and a range of \(\{y \in \mathbb{R}\}\). There are no x-intercepts, and the y-intercept is (0, -2). The graph has a vertical asymptote at x = 1 and a horizontal asymptote at y = 0. The graph is increasing and concave up for \(x < 1\), and decreasing and concave down for \(x > 1\).
1Step 1: Determine the Domain and Range
The domain of a function consists of all possible x-values. The only restriction for the domain of a rational function is that the denominator cannot be equal to zero. In our case, the denominator is \(x - 1\), so we have to find the value of x that makes it zero.
\[
x - 1 = 0 \implies x = 1
\]
Thus, the domain of g(x) is \(\{x \in \mathbb{R} | x \neq 1\}\).
To determine the range (all possible y-values), we need to find the vertical asymptote. By analyzing the denominator, it's clear that a vertical asymptote exists at x = 1. Since there is no limit on the y-values as x approaches the asymptote, the range of g(x) is all real numbers. So, the range is \(\{y \in \mathbb{R}\}\).
2Step 2: Determine the Intercepts
To find the x-intercept(s), we need to find the value(s) of x when g(x) = 0. However, since the numerator is not equal to 0 for any value of x, there are no x-intercepts.
To find the y-intercept, we need to find the value of g(x) when x = 0:
\[
g(0) = \frac{2}{0 - 1} = -2
\]
Thus, the y-intercept is the point (0, -2).
3Step 3: Find the Asymptotes
As we established earlier, g(x) has a vertical asymptote at x = 1.
To find the horizontal asymptote, we notice that the degree of the numerator is smaller than the degree of the denominator. In such cases, the horizontal asymptote is at y = 0.
4Step 4: Determine Intervals of Increase/Decrease, and Concavity
Calculate the first and second derivatives of the given function:
\[
g'(x) = -\frac{2}{(x - 1)^2};
\]
\[
g''(x) = \frac{4}{(x - 1)^3}
\]
Using the first derivative, g'(x), we can determine intervals where the function is increasing or decreasing:
- For \(x < 1\): \(g'(x) > 0\), the graph is increasing.
- For \(x > 1\): \(g'(x) < 0\), the graph is decreasing.
Using the second derivative, g''(x), we can determine intervals where the function is concave up or down:
- For \(x < 1\): \(g''(x) > 0\), the graph is concave up.
- For \(x > 1\): \(g''(x) < 0\), the graph is concave down.
With all this information, we can now sketch a representative graph for the given function \(g(x) = \frac{2}{x-1}\).
Key Concepts
Domain and RangeAsymptotesDerivativesConcavity
Domain and Range
In the world of curve sketching, understanding the domain and range of a function is crucial. They define where the function exists and the output values it can take. For the function \( g(x) = \frac{2}{x-1} \), the domain includes all real numbers except where the denominator is zero.
This happens when \( x = 1 \). So, the domain is every real number except 1: \( \{x \in \mathbb{R} | x eq 1\} \).
For the range, since the function is not a constant and covers all values except at the asymptote, the range is all real numbers: \( \{y \in \mathbb{R}\} \).
This happens when \( x = 1 \). So, the domain is every real number except 1: \( \{x \in \mathbb{R} | x eq 1\} \).
For the range, since the function is not a constant and covers all values except at the asymptote, the range is all real numbers: \( \{y \in \mathbb{R}\} \).
- Domain: All real numbers except \( x = 1 \)
- Range: All real numbers \( y \)
Asymptotes
Asymptotes are like invisible boundaries in a graph toward which the function heads but never really touches. They guide the overall shape of the graph.
For the function \( g(x) = \frac{2}{x-1} \), a vertical asymptote occurs at \( x = 1 \).
This is where the function becomes undefined, creating that natural boundary. For the horizontal asymptote, when the degree of the numerator is less than that of the denominator, like here, the x-axis (\( y = 0 \)) is the horizontal asymptote.
This tells us that as \( x \) moves away from 1 to infinity (positive or negative), the function approaches zero.
For the function \( g(x) = \frac{2}{x-1} \), a vertical asymptote occurs at \( x = 1 \).
This is where the function becomes undefined, creating that natural boundary. For the horizontal asymptote, when the degree of the numerator is less than that of the denominator, like here, the x-axis (\( y = 0 \)) is the horizontal asymptote.
This tells us that as \( x \) moves away from 1 to infinity (positive or negative), the function approaches zero.
- Vertical Asymptote: \( x = 1 \)
- Horizontal Asymptote: \( y = 0 \)
Derivatives
Derivatives are powerful tools to analyze how a function behaves over its domain.
The first derivative, \( g'(x) = -\frac{2}{(x - 1)^2} \), indicates the slope of \( g(x) \). It tells us where the curve is rising or falling. Since \( g'(x) > 0 \) for \( x < 1 \), the function increases. Similarly, \( g'(x) < 0 \) for \( x > 1 \), where the curve decreases.
Derivatives help in determining key characteristics like:
The first derivative, \( g'(x) = -\frac{2}{(x - 1)^2} \), indicates the slope of \( g(x) \). It tells us where the curve is rising or falling. Since \( g'(x) > 0 \) for \( x < 1 \), the function increases. Similarly, \( g'(x) < 0 \) for \( x > 1 \), where the curve decreases.
Derivatives help in determining key characteristics like:
- Increasing: \( x < 1 \)
- Decreasing: \( x > 1 \)
Concavity
Concavity describes the direction of the curve's bend. The second derivative, \( g''(x) = \frac{4}{(x - 1)^3} \), reveals whether the graph is cupped upwards or downwards.
When \( g''(x) > 0 \) for \( x < 1 \), the graph is concave up, resembling a smile. On the other hand, \( g''(x) < 0 \) for \( x > 1 \) indicates the graph is concave down, or a frown.Understanding concavity aids in comprehending the graph's overall shape:
When \( g''(x) > 0 \) for \( x < 1 \), the graph is concave up, resembling a smile. On the other hand, \( g''(x) < 0 \) for \( x > 1 \) indicates the graph is concave down, or a frown.Understanding concavity aids in comprehending the graph's overall shape:
- Concave Up: \( x < 1 \)
- Concave Down: \( x > 1 \)
Other exercises in this chapter
Problem 48
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