Problem 85
Question
The function \(f(x)=x^{6}\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?
Step-by-Step Solution
Verified Answer
By restricting the domain of the function to non-negative real numbers (i.e., \(x \geq 0\)), you can make the function \(f(x)=x^{6}\) one-to-one.
1Step 1: Understanding the given function
The provided function is \(f(x)=x^{6}\). The graph of this function is similar to a parabola's, except it's even steeper. The y-value is always positive or zero (for \(x=0\)), and for every positive y-value, there correspond both a positive and a negative x-value. This implies that the function is not one-to-one.
2Step 2: Figuring out domain restrictions
To make the function \(f(x)=x^{6}\) one-to-one, the best way is to restrict the domain so that it includes only non-negative numbers. In math terms, this means that the domain for function \(f(x)\) should be \(x\geq0\). With this domain restriction, for all x in the domain, there is exactly one unique y in the range of f. Similarly, for each y in the range of f, there is exactly one x in the domain.
Key Concepts
Understanding Domain RestrictionExploring Function TransformationInvestigating Even-Degree Polynomials
Understanding Domain Restriction
When we speak of domain restriction, we're addressing how to limit the set of inputs that a function can accept. This action can transform a function that is not one-to-one into one that has this desirable characteristic, meaning for every output, there is a unique input.
For the function in the exercise, \( f(x) = x^6 \), restricting the domain to non-negative numbers creates a one-to-one function. We do this by defining a new domain where \( x \geq 0 \). With this restriction, each input value (\( x \)) will have a unique output (\( f(x) \)), and the function will satisfy the horizontal line test, which is a visual method to determine if a function is one-to-one.
To implement a domain restriction, one must examine the behavior of the function and strategically choose a subset of the entire possible domain. This technique is commonly used to solve problems involving inverse functions and offers practical application in real-life scenarios where limitations on inputs are necessary for the desired outcomes.
For the function in the exercise, \( f(x) = x^6 \), restricting the domain to non-negative numbers creates a one-to-one function. We do this by defining a new domain where \( x \geq 0 \). With this restriction, each input value (\( x \)) will have a unique output (\( f(x) \)), and the function will satisfy the horizontal line test, which is a visual method to determine if a function is one-to-one.
To implement a domain restriction, one must examine the behavior of the function and strategically choose a subset of the entire possible domain. This technique is commonly used to solve problems involving inverse functions and offers practical application in real-life scenarios where limitations on inputs are necessary for the desired outcomes.
Exploring Function Transformation
The term function transformation refers to the various ways we can adjust a function’s graph. There are several types of transformations: translations, reflections, stretching, and compressing. These transformations can be applied to the original function to produce a desired effect on the graph.
Considering our function \( f(x) = x^6 \), suppose we want to center its graph at a different point or adjust its steepness, we could apply a vertical shift or a stretching/compressing transformation, respectively. If we, for instance, apply a reflection over the x-axis with \( g(x) = -f(x) \), we obtain a graph that goes downward as it moves away from the y-axis instead of upward.
Transformation tools are essential to get a better understanding of complex behavior by visualizing different aspects of functions, and they enable us to model real-world situations more accurately within a defined context.
Considering our function \( f(x) = x^6 \), suppose we want to center its graph at a different point or adjust its steepness, we could apply a vertical shift or a stretching/compressing transformation, respectively. If we, for instance, apply a reflection over the x-axis with \( g(x) = -f(x) \), we obtain a graph that goes downward as it moves away from the y-axis instead of upward.
Transformation tools are essential to get a better understanding of complex behavior by visualizing different aspects of functions, and they enable us to model real-world situations more accurately within a defined context.
Investigating Even-Degree Polynomials
Polynomials are algebraic expressions with one or more terms, where each term consists of a coefficient multiplied by a variable raised to a non-negative integer power. An even-degree polynomial is a specific type of polynomial where the largest exponent is an even number.
In our example, \( f(x) = x^6 \), the highest power of \( x \) is 6, which is an even number, so it is an even-degree polynomial. These kinds of polynomials have certain characteristics. They have end behavior that is the same in both directions: either both ends rise or both ends fall as \( x \) moves towards positive and negative infinity.
They can also have multiple real roots, complex roots, and can display a variety of behaviors in between, including having a minimum or maximum point. However, the simplest even-degree polynomials, like the one in our exercise, will always have a single minimum or maximum and this occurs at the vertex, the highest or lowest point on the graph, respectively. Understanding the nature of even-degree polynomials help us to predict the function's long-term behavior and how it reacts under domain restrictions and transformations.
In our example, \( f(x) = x^6 \), the highest power of \( x \) is 6, which is an even number, so it is an even-degree polynomial. These kinds of polynomials have certain characteristics. They have end behavior that is the same in both directions: either both ends rise or both ends fall as \( x \) moves towards positive and negative infinity.
They can also have multiple real roots, complex roots, and can display a variety of behaviors in between, including having a minimum or maximum point. However, the simplest even-degree polynomials, like the one in our exercise, will always have a single minimum or maximum and this occurs at the vertex, the highest or lowest point on the graph, respectively. Understanding the nature of even-degree polynomials help us to predict the function's long-term behavior and how it reacts under domain restrictions and transformations.
Other exercises in this chapter
Problem 84
Explain why the function \(f(x)=2^{x}\) has no vertical asymptotes (review Section 4.6).
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The function \(f(x)=|x+2|\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?
View solution