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Chapter 6: Problem 10

Let \(f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be the function defined by \(f(x, y)=-x^{2} y+3 y,\) for all \((x, y) \in \mathbb{R} \times \mathbb{R} .\) Is the function \(f\) an injection? Is the function \(f\) a surjection? Justify your conclusions.

Short Answer

Expert verified
In conclusion, the function \(f(x, y) = -x^{2}y + 3y\) is not injective, since we cannot prove that \(x_1 = x_2\) and \(y_1 = y_2\) for every \(f(x_1, y_1) = f(x_2, y_2)\). However, it is surjective, as for any \(z \in \mathbb{R}\), there exists \((x, y) \in \mathbb{R} \times \mathbb{R}\) such that \(f(x, y) = z\).

Step by step solution

01

Check for Injection

To check if the function \(f(x, y)=-x^2y + 3y\) is injective, we need to show that if \(f(x_1, y_1) = f(x_2, y_2)\), then \((x_1, y_1) = (x_2, y_2)\). Let's assume that \(f(x_1, y_1) = f(x_2, y_2)\) and try to show that \((x_1, y_1) = (x_2, y_2)\). So, we have \(-x_1^{2} y_1 + 3y_1 = -x_2^{2} y_2 + 3y_2\). Now we manipulate this equation to reach our conclusion: \( (-x_1^2 + 3) y_1 = (-x_2^2 + 3) y_2 \) If either \(y_1 = 0\) or \(y_2=0\), we can directly observe that \(y_1=y_2=0\) (since \(x^2\) only adds a positive value). So now let's consider \(y_1 \neq 0\) and \(y_2 \neq 0\): \( \frac{(-x_1^2 +3)}{y_1} = \frac{(-x_2^2 +3)}{y_2} \) \( \frac{x_1^2}{y_1} = \frac{x_2^2}{y_2} \) Notice that on both sides the values inside the fractions must be positive. But we're still missing a crucial part. We can multiply both sides by any positive constant \(c\) and still get the same result: \( \frac{c x_1^2}{c y_1} = \frac{c x_2^2}{c y_2} \) However, we cannot prove that \(x_1 = x_2\) and \(y_1 = y_2\). This indicates that the function is not injective.
02

Check for Surjection

To check if the function \(f(x, y)=-x^2y + 3y\) is surjective, we need to show that for every \(z \in \mathbb{R}\), there exists \((x, y) \in \mathbb{R} \times \mathbb{R}\) such that \(f(x, y) = z\). Given \(z \in \mathbb{R}\), our function has the form: \( f(x, y) = -x^2 y + 3y = z \) We are looking for \((x, y) \in \mathbb{R} \times \mathbb{R}\) that satisfy this equation. It's important to notice that when \(y = 0\), the function is always equal to 0 no matter the value of \(x\). Thus, we can represent any value \(z = 0\) just by choosing \(y=0\) and any \(x \in \mathbb{R}\). When \(y \neq 0\), we can try to represent the function in terms of a single variable: \(f(x, y) = y(-x^2 + 3) = z \) We can observe that for any \(z\), by choosing \(y=z\) and \(x=\sqrt{3}\), the last expression will be: \(f(\sqrt{3}, z) = z( -(\sqrt{3})^2 + 3) = z(-3 + 3) = 0 = z \) Thus, for any \(z \in \mathbb{R}\), there exists \((x, y) \in \mathbb{R} \times \mathbb{R}\) such that \(f(x, y) = z\), which means that the function is surjective. #Conclusion# The function \(f(x, y)=-x^2y + 3y\) is not injective but it is surjective.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Analysis
In mathematics, function analysis involves determining the behaviors and characteristics of functions. Often, we look at properties like injectivity, surjectivity, and bijectivity. These properties tell us how inputs relate to outputs in a function and help us understand the nature of different kinds of functions.

For injectivity (or one-to-one function), every element of the function's codomain should map back to at most one element of its domain. This means, essentially, that different inputs always lead to different outputs. If two different input pairs
  • Injective: Consider the equation \(-x_1^2 y_1 + 3y_1 = -x_2^2 y_2 + 3y_2\). To prove injectivity, one must be able to show that if this equation holds, \(x_1 = x_2\) and \(y_1 = y_2\). Our function \(f(x, y) = -x^2 y + 3y\) was shown to lack this property, as we couldn't uniquely solve for \(x_1 = x_2\) and \(y_1 = y_2\) solely from the output.
  • Surjective: Every element in the codomain is an output for some input in the domain. In simpler terms, all possible outputs can be achieved by plugging in some input into the function. For this solution, we examined \(-x^2 y + 3y = z\) and showed that every real number \(z\) can be obtained by choosing appropriate \(x\) and \(y\). Thus, our function is considered surjective.
Real-Valued Functions
A real-valued function is a function that assigns a real number to each member of its domain. In the context of our exercise, the function \(f(x, y) = -x^2 y + 3y\) assigns a real number to each ordered pair \((x, y)\) in the real plane \(\mathbb{R} \times \mathbb{R}\). These types of functions are widespread in mathematics and greatly useful in real-world applications because they model real phenomena.

Understanding real-valued functions involves exploring what happens when parameters change.
  • Domain and Codomain: Here, the domain is \(\mathbb{R} \times \mathbb{R}\) and the codomain is \(\mathbb{R}\). This indicates that both inputs are real numbers, and every calculation results in a real number.
  • Behavior and Graphing: The behavior of a function like \(-x^2 y + 3y\) can be studied by graphing. For instance, by varying \(x\) and \(y\) values, we can visualize how the function behaves and where it reaches certain values.
Properties of Functions
When studying functions, several fundamental properties come into play. Knowing these characteristics enables us to understand how functions behave under different conditions and constraints.

A few essential properties widely analyzed in function study are:
  • Injectivity: As mentioned earlier, a function is injective if distinct inputs lead to distinct outputs. While assessing our function, we found it does not satisfy injectivity, as the condition couldn't be verified.
  • Surjectivity: This property ensures that every possible output is attainable by some input, which our function demonstrated effectively by choosing appropriate \(x\) and \(y\) pairs.
  • Bijectivity: If a function is both injective and surjective, it is called bijective. This property would mean the function has a perfect one-to-one correspondence between elements of its domain and codomain, which is not the case for our exercise function as it isn't injective.
Understanding these properties lets us precisely determine how a function interacts within its defined space, enabling further applications in calculus, algebra, and even disciplines beyond mathematics.

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Most popular questions from this chapter

The number of divisors function. Let \(d\) be the function that associates with each natural number the number of its natural number divisors. That is, \(d: \mathbb{N} \rightarrow \mathbb{N}\) where \(d(n)\) is the number of natural number divisors of \(n\). For example, \(d(6)=4\) since \(1,2,3,\) and 6 are the natural number divisors of 6 (a) Calculate \(d(k)\) for each natural number \(k\) from 1 through 12 . (b) Does there exist a natural number \(n\) such that \(d(n)=1 ?\) What is the set of preimages of the natural number \(1 ?\) (c) Does there exist a natural number \(n\) such that \(d(n)=2 ?\) If so, determine the set of all preimages of the natural number \(2 .\) (d) Is the following statement true or false? Justify your conclusion. For all \(m, n \in \mathbb{N},\) if \(m \neq n,\) then \(d(m) \neq d(n) .\) (e) Calculate \(d\left(2^{k}\right)\) for \(k=0\) and for each natural number \(k\) from 1 through 6 (f) Based on your work in Exercise (6e), make a conjecture for a formula for \(d\left(2^{n}\right)\) where \(n\) is a nonnegative integer. Then explain why your conjecture is correct. (g) Is the following statement is true or false? For each \(n \in \mathbb{N},\) there exists a natural number \(m\) such that \(d(m)=n\)

Let \(D=\mathbb{N}-\\{1,2\\}\) and define \(d: D \rightarrow \mathbb{N} \cup\\{0\\}\) by \(d(n)=\) the number of diagonals of a convex polygon with \(n\) sides. In Preview Activity \(1,\) we showed that for values of \(n\) from 3 through 8 , $$ d(n)=\frac{n(n-3)}{2} $$ Use mathematical induction to prove that for all \(n \in D\), $$ d(n)=\frac{n(n-3)}{2} $$

Let \(C\) be the set of all real functions that are continuous on the closed interval \([0,1] .\) Define the function \(A: C \rightarrow \mathbb{R}\) as follows: For each \(f \in C,\) $$A(f)=\int_{0}^{1} f(x) d x$$ Is the function \(A\) an injection? Is it a surjection? Justify your conclusions.

In Exercise (6), we introduced the number of divisors function \(d\). For this function, \(d: \mathbb{N} \rightarrow \mathbb{N},\) where \(d(n)\) is the number of natural number divisors of \(n\) A function that is related to this function is the so-called set of divisors function. This can be defined as a function \(S\) that associates with each natural number the set of its distinct natural number factors. For example. \(S(6)=\\{1,2,3,6\\}\) and \(S(10)=\\{1,2,5,10\\}\) (a) Discuss the function \(S\) by carefully stating its domain, codomain, and its rule for determining outputs. (b) Determine \(S(n)\) for at least five different values of \(n\). - (c) Determine \(S(n)\) for at least three different prime number values of \(n\). (d) Does there exist a natural number \(n\) such that card \((S(n))=1 ? \mathrm{Ex}-\) plain. [Recall that card \((S(n))\) is the number of elements in the set \(S(n) .]\) (e) Does there exist a natural number \(n\) such that card \((S(n))=2 ? \mathrm{Ex}=\) plain. (f) Write the output for the function \(d\) in terms of the output for the function \(S\). That is, write \(d(n)\) in terms of \(S(n)\). (g) Is the following statement true or false? Justify your conclusion. For all natural numbers \(m\) and \(n,\) if \(m \neq n,\) then \(S(m) \neq S(n)\). (h) Is the following statement true or false? Justify your conclusion. For all sets \(T\) that are subsets of \(\mathbb{N}\), there exists a natural number \(n\) such that \(S(n)=T\)

For each of the following functions, determine if the function is an injection and determine if the function is a surjection. Justify all conclusions. (a) \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f(x)=3 x+1,\) for all \(x \in \mathbb{Z}\). (b) \(F: \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(F(x)=3 x+1,\) for all \(x \in \mathbb{Q}\) (c) \(g: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(g(x)=x^{3},\) for all \(x \in \mathbb{R}\) (d) \(G: \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(G(x)=x^{3},\) for all \(x \in \mathbb{Q}\). (e) \(k: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(k(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}\) (f) \(K: \mathbb{R}^{*} \rightarrow \mathbb{R}\) defined by \(K(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}^{*}\). Note: \(\mathbb{R}^{*}=\\{x \in \mathbb{R} \mid x \geq 0\\}\). (g) \(K_{1}: \mathbb{R}^{*} \rightarrow T\) defined by \(K_{1}(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}^{*},\) where \(T=\) \(\\{y \in \mathbb{R} \mid 0
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