Problem 103
Question
(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=\frac{3 x^{2}}{x^{2}+1}$$
Step-by-Step Solution
Verified Answer
The inverse relation of the given function is not a function. This conclusion was reached by applying the horizontal line test on the graph of the inverse relation, which showed that there were points where a horizontal line intersected the graph more than once, implying that there's more than one output for a single input in the inverse relation.
1Step 1: Graph the function
Using a graphing utility, graph the function \(f(x)=\frac{3 x^{2}}{x^{2}+1}\). The graph shows a curve that tends to flatten as the function approaches to positive infinity.
2Step 2: Draw the inverse relation
By switching the roles of y and x in the function equation to \(x = \frac{3 y^{2}}{y^{2}+1}\), the graphing utility can then be used to draw the inverse relation which results in a curve that is a reflection about y = x line.
3Step 3: Determine if the inverse relation is a function
A relation is a function if every input corresponds to exactly one output. By using the horizontal line test, which states that if a horizontal line intersects the graph in more than one point, then the relation is not a function. From the graph of the inverse relation, it's clear that there are points where a horizontal line intersects the curve more than once, hence the inverse relation is not a function.
Key Concepts
Graphing UtilitiesHorizontal Line TestInverse Relations
Graphing Utilities
Graphing utilities are essential tools for visualizing mathematical functions. These utilities can be software programs or calculators that produce detailed graphical representations of equations. This visual aid makes it easier to understand the behavior and properties of complex functions.
To graph a function using a graphing utility, you need to input the equation into the program. For the function \(f(x) = \frac{3 x^{2}}{x^{2}+1}\), the graphing tool will plot the curve, showing how it behaves across different values of \(x\). As observed in the original exercise, the graph tends to flatten as \(x\) approaches positive infinity. This suggests that the function reaches a horizontal asymptote, indicating that the output values stabilize.
Using graphing utilities, students not only visualize the function but also analyze its behavior, make predictions, and confirm their calculations with ease.
To graph a function using a graphing utility, you need to input the equation into the program. For the function \(f(x) = \frac{3 x^{2}}{x^{2}+1}\), the graphing tool will plot the curve, showing how it behaves across different values of \(x\). As observed in the original exercise, the graph tends to flatten as \(x\) approaches positive infinity. This suggests that the function reaches a horizontal asymptote, indicating that the output values stabilize.
Using graphing utilities, students not only visualize the function but also analyze its behavior, make predictions, and confirm their calculations with ease.
Horizontal Line Test
The horizontal line test is a simple method to determine if a relation is a function—specifically, if an inverse relation qualifies as an inverse function.
A function's inverse, typically represented by switching the \(x\) and \(y\) values, is only considered an inverse function if every horizontal line intersects the graph at most once. This requirement stems from the definition of a function where each input should map to exactly one output.
For instance, when assessing the inverse relation of \(f(x) = \frac{3 x^{2}}{x^{2}+1}\), the graph of the inverse relation can be tested using horizontal lines. If a horizontal line crosses the graph at multiple points, it indicates multiple outputs for a single input, disproving its qualification as a function. As noted in the original solution, the inverse relation of this function fails the horizontal line test, meaning it is not an inverse function.
A function's inverse, typically represented by switching the \(x\) and \(y\) values, is only considered an inverse function if every horizontal line intersects the graph at most once. This requirement stems from the definition of a function where each input should map to exactly one output.
For instance, when assessing the inverse relation of \(f(x) = \frac{3 x^{2}}{x^{2}+1}\), the graph of the inverse relation can be tested using horizontal lines. If a horizontal line crosses the graph at multiple points, it indicates multiple outputs for a single input, disproving its qualification as a function. As noted in the original solution, the inverse relation of this function fails the horizontal line test, meaning it is not an inverse function.
Inverse Relations
Inverse relations are expressions where the roles of input and output are exchanged. If the original function is \(f(x)\), the inverse relation is generally expressed by switching \(x\) and \(y\) in the equation to solve for \(y\).
The original exercise demonstrates this by changing \(f(x) = \frac{3 x^{2}}{x^{2}+1}\) into \(x = \frac{3 y^{2}}{y^{2}+1}\). The inverse relation is portrayed visually as a reflection of the original function across the line \(y = x\).
While an inverse relation connects the outputs of a function back to inputs, it's crucial to determine if it is also a function. This would mean every \(x\) corresponds to exactly one \(y\). By using the horizontal line test, we can assert this quality. If the relation fails this test, then it isn't an inverse function but remains an inverse relation, showing a general correspondence between inputs and outputs of the initial function.
The original exercise demonstrates this by changing \(f(x) = \frac{3 x^{2}}{x^{2}+1}\) into \(x = \frac{3 y^{2}}{y^{2}+1}\). The inverse relation is portrayed visually as a reflection of the original function across the line \(y = x\).
While an inverse relation connects the outputs of a function back to inputs, it's crucial to determine if it is also a function. This would mean every \(x\) corresponds to exactly one \(y\). By using the horizontal line test, we can assert this quality. If the relation fails this test, then it isn't an inverse function but remains an inverse relation, showing a general correspondence between inputs and outputs of the initial function.
Other exercises in this chapter
Problem 102
(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, a
View solution Problem 102
Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture
View solution Problem 103
Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture
View solution Problem 104
(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, a
View solution