Problem 48
Question
Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$g(x)=\frac{4-x}{6 x^{2}}$$
Step-by-Step Solution
Verified Answer
To determine whether or not the function \(g(x) = \frac{4-x}{6x^2}\) is one-to-one and hence having an inverse, we need to graph the function and apply the Horizontal Line Test. If no horizontal line intersects the graph of the function at more than one point, then the function is one-to-one.
1Step 1: Understand the function
Firstly, we need to make sense of the function \(g(x) = \frac{4-x}{6x^2}\), which is a simple rational function where the numerator is \(4-x\) and the denominator is \(6x^2\).
2Step 2: Graph the function
Next thing to do is to graph the function \(g(x) = \frac{4-x}{6x^2}\) using a graphing utility. Do this by inputting the function in the graphing tool. The result should be a curve with the general shape of a rational function, which means it does not look like a straight line, but rather is a curve that approaches exclusive lines. The exact shape will depend on the specific graphing tool used.
3Step 3: Apply the Horizontal Line Test
After graphing the function, we must apply the Horizontal Line Test to the graph. Do this by imagining or drawing horizontal lines and discern whether any one line intersects the curve at more than one point. If the drawn horizontal line intersects at more than one point, then the function fails the Horizontal Line Test, hence the function is not one-to-one.
Key Concepts
Horizontal Line TestOne-to-One FunctionGraphing Rational Functions
Horizontal Line Test
The Horizontal Line Test is a simple visual check to determine whether a function is a one-to-one function. A function is considered to be one-to-one if no horizontal line intersects its graph at more than one point. This test helps us understand whether a function has an inverse that is also a function.
To perform the test, first graph the function you are investigating, such as the rational function given in the problem: \(g(x) = \frac{4-x}{6x^2}\). Once graphed, imagine a series of horizontal lines sweeping across the graph from top to bottom. If any line meets the curve in more than one place, the function is not one-to-one. In our example, visually inspecting the graph or using a graphing utility may show that multiple horizontal lines will intersect \(g(x)\) at more than one point.
Consequently, this would imply that the function does not have an inverse that is a function. This visual approach simplifies deciding whether or not a function meets the necessary criteria for having an inverse.
To perform the test, first graph the function you are investigating, such as the rational function given in the problem: \(g(x) = \frac{4-x}{6x^2}\). Once graphed, imagine a series of horizontal lines sweeping across the graph from top to bottom. If any line meets the curve in more than one place, the function is not one-to-one. In our example, visually inspecting the graph or using a graphing utility may show that multiple horizontal lines will intersect \(g(x)\) at more than one point.
Consequently, this would imply that the function does not have an inverse that is a function. This visual approach simplifies deciding whether or not a function meets the necessary criteria for having an inverse.
One-to-One Function
A one-to-one function is a function in which each element of the range is paired with exactly one element of the domain. This means that different input values will always result in different output values. Such a property ensures that the function has an inverse, and importantly, this inverse is also a function.
In algebraic terms, a function \(f\) is one-to-one if and only if \(f(a) = f(b)\) implies \(a = b\). This unique pair assignment setting can also be verified using the Horizontal Line Test as mentioned. If no horizontal line crosses the graph more than once, then the function is one-to-one.
In cases of rational functions, like \(g(x) = \frac{4-x}{6x^2}\), due to their complex nature, determining one-to-one correspondence might require using both algebraic methods and visual tests like the Horizontal Line Test.
In algebraic terms, a function \(f\) is one-to-one if and only if \(f(a) = f(b)\) implies \(a = b\). This unique pair assignment setting can also be verified using the Horizontal Line Test as mentioned. If no horizontal line crosses the graph more than once, then the function is one-to-one.
In cases of rational functions, like \(g(x) = \frac{4-x}{6x^2}\), due to their complex nature, determining one-to-one correspondence might require using both algebraic methods and visual tests like the Horizontal Line Test.
Graphing Rational Functions
Graphing rational functions involves understanding the behavior and characteristics of the function given by a ratio of polynomials. For the example \(g(x) = \frac{4-x}{6x^2}\), graphing is pivotal in visualizing key attributes like asymptotes, intercepts, and intervals of increase or decrease.
The use of a graphing utility can assist in quickly sketching these graphs, but it's beneficial to understand the underlying changes. Notably, rational functions often have vertical asymptotes at points where the denominator is zero, and horizontal or oblique asymptotes that indicate the end behavior of the graph as \(x\) approaches infinity or negative infinity.
For \(g(x)\), the graph will show a curve that avoids certain x-values (where \(6x^2 = 0\)) resulting in vertical asymptotes, and approaches a level line or another direction to signal horizontal or oblique asymptotes. Noticing these characteristics aids in understanding how rational functions behave, allowing for a better analysis of their properties, such as whether they are one-to-one.
The use of a graphing utility can assist in quickly sketching these graphs, but it's beneficial to understand the underlying changes. Notably, rational functions often have vertical asymptotes at points where the denominator is zero, and horizontal or oblique asymptotes that indicate the end behavior of the graph as \(x\) approaches infinity or negative infinity.
For \(g(x)\), the graph will show a curve that avoids certain x-values (where \(6x^2 = 0\)) resulting in vertical asymptotes, and approaches a level line or another direction to signal horizontal or oblique asymptotes. Noticing these characteristics aids in understanding how rational functions behave, allowing for a better analysis of their properties, such as whether they are one-to-one.
Other exercises in this chapter
Problem 47
(a) find the slope and y-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify yo
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Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of \(g\) and \(h\) relative to the graph of \(f\).$$\begin{a
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Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=x^{1 / 4}, \quad g(x)=x^{4}$$
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Sketch the graph of the function by hand. Then use a graphing utility to verify the graph. $$f(x)=[[x]]-3$$
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