Problem 82
Question
Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=(x-1)^{3}$$
Step-by-Step Solution
Verified Answer
Yes, the function \(f(x) = (x-1)^3\) is one-to-one, thus it does have an inverse that is also a function.
1Step 1: Graph the given function
The function given is \(f(x) = (x-1)^3\). We begin by graphing the function using a graphing utility or tool. This function is a cubic function that has been shifted one unit to the right. It shapes like a cubic curve, opening upwards and downwards, and passes through the point (1,0).
2Step 2: Perform the horizontal line test
Once we have the graph, we can check if the function is one-to-one by using the horizontal line test. A function is one-to-one if every horizontal line intersects the graph in at most one point. When applying the test to our graph, it can be seen that every horizontal line touches the graph at exactly one point.
3Step 3: Determining the result
According to the horizontal line test, the given function is one-to-one because it passes the test. This suggests that an inverse of the function exists and is indeed a function.
Key Concepts
Understanding One-to-One FunctionsUtilizing Graphing UtilitiesSimplifying with the Horizontal Line Test
Understanding One-to-One Functions
A one-to-one function is a type of function where each output value is paired with exactly one input value. This means that no two inputs will give the same output. This uniqueness is what ensures the function has an inverse that is also a function.
Why is this important? If a function is not one-to-one, its inverse will not be a function. For example, if two inputs result in the same output, the inverse would not know which input to pair with that output, causing confusion!
Think of one-to-one functions as a secure lock-and-key system, where each lock (input) has exactly one unique key (output). This exclusivity is the essence of one-to-one functions and perfectly paves the way for inverses.
Why is this important? If a function is not one-to-one, its inverse will not be a function. For example, if two inputs result in the same output, the inverse would not know which input to pair with that output, causing confusion!
Think of one-to-one functions as a secure lock-and-key system, where each lock (input) has exactly one unique key (output). This exclusivity is the essence of one-to-one functions and perfectly paves the way for inverses.
Utilizing Graphing Utilities
Graphing utilities are powerful tools that help visualize functions and understand their properties. They can include software applications, online tools, or graphing calculators. By using these, complex functions become far easier to analyze and comprehend.
With graphing utilities, you can plot the graph of the function, observe its shape, and gain insights into its behavior. For the given function, \( f(x) = (x-1)^3 \), graphing it helps you see the distinct cubic curve.
This visual aid not only simplifies the understanding of one-to-one functions but also assists when applying tests like the horizontal line test. Remember, a picture is worth a thousand equations!
With graphing utilities, you can plot the graph of the function, observe its shape, and gain insights into its behavior. For the given function, \( f(x) = (x-1)^3 \), graphing it helps you see the distinct cubic curve.
This visual aid not only simplifies the understanding of one-to-one functions but also assists when applying tests like the horizontal line test. Remember, a picture is worth a thousand equations!
Simplifying with the Horizontal Line Test
The horizontal line test is a straightforward method for determining if a function is one-to-one. By drawing horizontal lines across the graph of the function, you can check if each line intersects the graph at only one point.
If every line meets the graph once, the function is one-to-one; if any line meets it at more than one point, the function fails the test.
This test is immensely helpful because it turns a potentially complex algebraic check into an easy visual test.
Viewing the function \( f(x) = (x-1)^3 \), passing the horizontal line test confirms that it is a one-to-one function, ensuring an inverse exists. This makes analysis faster and often more intuitive.
If every line meets the graph once, the function is one-to-one; if any line meets it at more than one point, the function fails the test.
This test is immensely helpful because it turns a potentially complex algebraic check into an easy visual test.
Viewing the function \( f(x) = (x-1)^3 \), passing the horizontal line test confirms that it is a one-to-one function, ensuring an inverse exists. This makes analysis faster and often more intuitive.
Other exercises in this chapter
Problem 81
Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=|x|+4$$
View solution Problem 82
Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=-x^{2}-3 x+1$$
View solution Problem 82
Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=\frac{1}{4 x+5}$$
View solution Problem 82
Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=|x|+3$$
View solution