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Chapter 5: Problem 2

Does the formula \(f(x)=1 /\left(x^{2}-2\right)\) define a function \(f: \mathbf{R} \rightarrow \mathbf{R}\) ? A function \(f: \mathbf{Z} \rightarrow \mathbf{R}\) ?

Short Answer

Expert verified
\(\)The formula \(f(x)\) does not define a function \(f: \mathbf{R} \rightarrow \mathbf{R}\) because \(f(x)\) is undefined for \(x=\sqrt{2}\) and \(x=-\sqrt{2}\). It, however, defines a function \(f: \mathbf{Z} \rightarrow \mathbf{R}\) because any integer can be an input for \(f(x)\).

Step by step solution

01

Analysis of the formula

Analyze \(f(x)=1 /\left(x^{2}-2\right)\) for all complex numbers. It is seen that \(f(x)\) is undefined if the denominator is zero. So, solve \(x^{2}-2=0\) to find for which values of \(x\), \(f(x)\) is undefined.
02

Finding Undefined Values

Solving \(x^{2}-2=0\), we get \(x=\sqrt{2}\) and \(x=-\sqrt{2}\). Therefore, \(f(x)\) is undefined for \(x=\sqrt{2}\) and \(x=-\sqrt{2}\).
03

Determine if \(f: \mathbf{R} \rightarrow \mathbf{R}\)

The set \(\mathbf{R}\) represents all real numbers. Since \(f(x)\) is undefined for \(x=\sqrt{2}\) and \(x=-\sqrt{2}\), all real numbers cannot be inputs for \(f(x)\). Therefore, \(f(x)\) doesn't represent a function \(\mathbf{R} \rightarrow \mathbf{R}\)
04

Determine if \(f: \mathbf{Z} \rightarrow \mathbf{R}\)

The set \(\mathbf{Z}\) represents all integers. Since \(\sqrt{2}\) and \(-\sqrt{2}\) are not integers, any integer can be an input for \(f(x)\), therefore \(f(x)\) does represent a function \(f: \mathbf{Z} \rightarrow \mathbf{R}\)

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Most popular questions from this chapter

For \(n \in \mathbf{Z}^{+}\), define \(\tau: \mathbf{Z}^{+} \rightarrow \mathbf{Z}^{+}\)by \(\tau(n)=\) the number of positive-integer divisors of \(n\). a) Let \(n=p_{1}^{e_{1}} p_{2}^{\epsilon_{2}} p_{3}^{e_{3}} \ldots p_{k}^{e_{k}}\), where \(p_{1}, p_{2}, p_{3}, \ldots, p_{k}\) are distinct primes and \(e_{t}\) is a positive integer for all \(1 \leq i \leq k\). What is \(\tau(n)\) ? b) Determine the three smallest values of \(n \in \mathbf{Z}^{+}\)for which \(\tau(n)=k\), where \(k=2,3,4,5,6\). c) For all \(k \in \mathbf{Z}^{+}, k>1\), prove that \(\tau^{-1}(k)\) is infinite. d) If \(a, b \in \mathbf{Z}^{+}\)with \(\operatorname{gcd}(a, b)=1\), prove that \(\tau(a b)=\) \(\tau(a) \tau(b)\)

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be the integer sequence defined recursively by 1) \(a_{1}=0\); and 2) For \(n>1, a_{n}=1+a_{[n / 2\rfloor}\). Prove that \(a_{n}=\left\lfloor\log _{2} n\right\rfloor\) for all \(n \in \mathbf{Z}^{+}\).

Let \(A=\\{x, a, b, c, d\\}\). a) How many closed binary operations \(f\) on \(A\) satisfy \(f(a, b)=c ?\) b) How many of the functions \(f\) in part (a) have \(x\) as an identity? c) How many of the functions \(f\) in part (a) have an identity? d) How many of the functions \(f\) in part (c) are commutative?

Determine all real numbers \(x\) for which $$ x^{2}-\lfloor x\rfloor=1 / 2 $$

At St. Xavier High School ten candidates \(C_{1}, C_{2}, \ldots, C_{10}\), run for senior class president. a) How many outcomes are possible where (i) there are no ties (that is, no two, or more, candidates receive the same number of votes? (ii) ties are permitted? [Here we may have an outcome such as \(\left\\{C_{2}, C_{3}, C_{7}\right\\},\left\\{C_{1}, C_{4}, C_{9}, C_{10}\right\\}\), \(\left\\{C_{5}\right\\},\left\\{C_{6}, C_{8}\right\\}\), where \(C_{2}, C_{3}, C_{7}\) tie for first place, \(C_{1}, C_{4}, C_{9}, C_{10}\) tie for fourth place, \(C_{5}\) is in eighth place, and \(C_{6}, C_{8}\) are tied for ninth place.] (iii) three candidates tie for first place (and other ties are permitted)? b) How many of the outcomes in section (iii) of part (a) have \(C_{3}\) as one of the first-place candidates? c) How many outcomes have \(C_{3}\) in first place (alone, or tied with others)?
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