Problem 35
Question
Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=\frac{4 x-4}{x^{2}}\) Domain: \((-\infty, 0) \cup(0, \infty)\) Intercept: \(x\) -intercept: 1 Asymptotes: \(x\) -axis and \(y\) -axis Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((0,2)\); \(\searrow\) on \((-\infty, 0) \cup(2, \infty)\) Relative extrema: Rel. max. at \((2,1)\) Concavity: Downward on \((-\infty, 0) \cup(0,3)\); upward on \((3, \infty)\) Point of inflection: \(\left(3, \frac{8}{9}\right)\)
Step-by-Step Solution
Verified Answer
To sketch the graph of \(f(x)=\frac{4x-4}{x^2}\), plot the domain on the x-axis: (-∞, 0) ∪ (0, ∞). Plot the x-intercept (1,0) and draw asymptotes at x=0 and y=0. The function is increasing on (0,2) and decreasing on (-∞, 0) ∪ (2, ∞). Plot the relative maximum at (2,1) and the inflection point at (3, 8/9). Follow the concavity information to draw the curve of the function.
1Step 1: Plot the Domain of the Function
The function is defined for x values in the domain, which is given by (-∞, 0) ∪ (0, ∞). Plot the domain on the x-axis.
2Step 2: Plot the x-intercept
The function intercepts the x-axis at x=1. Plot this point (1,0) on the graph.
3Step 3: Draw Asymptotes
The function has asymptotes on the x-axis and the y-axis. Draw a horizontal dashed line on the x-axis (y=0) and a vertical dashed line on the y-axis (x=0).
4Step 4: Determine and Plot Intervals of Increase and Decrease
The function is increasing on the interval (0,2) and decreasing on the intervals (-∞, 0) ∪ (2, ∞). Plot arrows on the graph to show the direction of the function within these intervals.
5Step 5: Plot Relative Extrema
There is a relative maximum at the point (2,1). Plot this point on the graph and label it as a maximum.
6Step 6: Determine and Plot Concavity and Inflection Points
The function is concave down on the intervals (-∞, 0) ∪ (0,3) and concave up on the interval (3, ∞). The inflection point is given at (3, 8/9). Plot the inflection point on the graph and draw the curve of the function following the concavity information.
Now you can see the sketch of the graph of the function \(f(x)=\frac{4x-4}{x^2}\) based on the given information.
Key Concepts
Domain and RangeAsymptotesRelative ExtremaConcavity and Inflection Points
Domain and Range
Understanding the domain and range of a rational function is essential for graphing it correctly. The domain of a function includes all the potential inputs (x-values) where the function is defined. For the function \(f(x)=\frac{4x-4}{x^2}\), the domain is \((-\infty, 0) \cup (0, \infty)\). This means the function is defined for all real numbers except zero. The value of \(x\) cannot be zero because it would make the denominator zero, causing the function to be undefined.
The range, on the other hand, represents all possible output values (y-values) of the function. Since \(f(x)=\frac{4x-4}{x^2}\), this implies \(f(x)\) can take any real number value except zero. Thus, the range is also \((-\infty, 0) \cup (0, \infty)\), meaning \(f(x)\) never actually attains zero, even though it can get arbitrarily close.
Identifying these sets is crucial when sketching the function, as they dictate which portions of the x-axis (and therefore the graph) you should focus on.
The range, on the other hand, represents all possible output values (y-values) of the function. Since \(f(x)=\frac{4x-4}{x^2}\), this implies \(f(x)\) can take any real number value except zero. Thus, the range is also \((-\infty, 0) \cup (0, \infty)\), meaning \(f(x)\) never actually attains zero, even though it can get arbitrarily close.
Identifying these sets is crucial when sketching the function, as they dictate which portions of the x-axis (and therefore the graph) you should focus on.
Asymptotes
Asymptotes are lines that a graph of a function approaches but never touches. They can be horizontal, vertical, or even oblique, based on the function. A rational function like \(f(x)=\frac{4x-4}{x^2}\) often includes both vertical and horizontal asymptotes.
In our case, the vertical asymptote occurs where the denominator equals zero, which is at \(x=0\). So, the y-axis acts as a barrier that the graph will approach but never cross. The function also has a horizontal asymptote along the x-axis, indicated by \(y=0\). This comes from the leading term of the polynomial in the numerator having a lesser degree than that in the denominator.
Visualizing these asymptotes as invisible boundaries helps in understanding how the graph behaves as it stretches towards infinity. On a plot, you might depict these with dashed lines to emphasize that the function never actually meets these lines, only gets asymptotically close.
In our case, the vertical asymptote occurs where the denominator equals zero, which is at \(x=0\). So, the y-axis acts as a barrier that the graph will approach but never cross. The function also has a horizontal asymptote along the x-axis, indicated by \(y=0\). This comes from the leading term of the polynomial in the numerator having a lesser degree than that in the denominator.
Visualizing these asymptotes as invisible boundaries helps in understanding how the graph behaves as it stretches towards infinity. On a plot, you might depict these with dashed lines to emphasize that the function never actually meets these lines, only gets asymptotically close.
Relative Extrema
Relative extrema refer to either relative maxima or minima where the function reaches a peak or a trough. For the function \(f(x)=\frac{4x-4}{x^2}\), these points show where the function changes its direction from increasing to decreasing, and vice versa.
This specific function has a relative maximum at the point \((2, 1)\). At this x-value, the function has its highest output compared to the surrounding values. It’s an essential feature to plot on the graph, as it indicates a turning point.
Identifying the relative extrema helps in sketching the graph because they tell us about the local "hills" or "valleys" along the graph. By marking these key points, you can better understand the graph's shape and direction, especially when paired with information about intervals of increase and decrease.
This specific function has a relative maximum at the point \((2, 1)\). At this x-value, the function has its highest output compared to the surrounding values. It’s an essential feature to plot on the graph, as it indicates a turning point.
Identifying the relative extrema helps in sketching the graph because they tell us about the local "hills" or "valleys" along the graph. By marking these key points, you can better understand the graph's shape and direction, especially when paired with information about intervals of increase and decrease.
Concavity and Inflection Points
Determining concavity involves understanding how a function curves, whether it has a "cup up" or "cup down" appearance. A function is concave downward if its graph curves down like an upside-down cup, and concave upward if it resembles a cup.
For \(f(x)=\frac{4x-4}{x^2}\), the function is concave downward on the intervals \((-\infty, 0) \cup (0, 3)\) and concave upward on \((3, \infty)\). This tells us how the slope or steepness of the function changes over different parts of the domain.
Inflection points, like at \((3, \frac{8}{9})\), signify where the graph changes its concavity. It’s the point where a "cup up" becomes a "cup down", or vice versa. Graphically marking these inflection points helps to precisely pinpoint where the graph's curvature policy shifts. By fully grasping the concepts of concavity and inflection, you gain deeper insight into the complex nature of a function's graph.
For \(f(x)=\frac{4x-4}{x^2}\), the function is concave downward on the intervals \((-\infty, 0) \cup (0, 3)\) and concave upward on \((3, \infty)\). This tells us how the slope or steepness of the function changes over different parts of the domain.
Inflection points, like at \((3, \frac{8}{9})\), signify where the graph changes its concavity. It’s the point where a "cup up" becomes a "cup down", or vice versa. Graphically marking these inflection points helps to precisely pinpoint where the graph's curvature policy shifts. By fully grasping the concepts of concavity and inflection, you gain deeper insight into the complex nature of a function's graph.
Other exercises in this chapter
Problem 34
Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(x)=x \sqrt{x+1} $$
View solution Problem 35
Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=e^{-x^{2}} \text { on }[-1,1] $$
View solution Problem 35
Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{1}{x-2} $$
View solution Problem 35
Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{2} e^{-x} $$
View solution