Problem 1
Graph the functions (a) \(y=16+2 x\) (b) \(y=8-2 x\) (c) \(y=2 x+12\) (in each case, consider the domain as consisting of nonnegative real numbers only.)
Problem 1
Given \(S_{1}=\\{3,6,9\\}, S_{2}=\\{a, b\\},\) and \(S_{3}=\\{m, m\\},\) find the Cartesian products: (a) \(S_{1} \times S_{2}\) \((b) S_{2} \times S_{3}\) (c) \(S_{3} \times S_{1}\)
Problem 1
Write the following in set notation: (a) The set of all real numbers greater than 34 (b) The set of ail real numbers greater than 8 but less than 65
Problem 2
Given the sets \(S_{1}=\\{2,4,6\\}, S_{2}=\\{7,2,6\\}, S_{3}=\\{4,2,6\\},\) and \(S_{4}=\\{2,4\\},\) which of the following statements are true? (a) \(S_{1}=S_{3}\) (b) \(S_{1}=R\) (set of real numbers) (c) \(8 \in S_{2}\) \((d) 3 \notin S_{2}\) \((e) 4 \notin S_{3}\) \((f) S_{4} \subset R\) \((g) S_{1} \supset S_{4}\) \((h) \varnothing \subset S_{2}\) (i) \(S_{3}=\\{1,2\\}\)
Problem 3
Graph the functions \((a) y=-x^{2}+5 x-2\) (b) \(y=x^{2}+5 x-2\) with the set of values \(-5 \leq x \leq 5\) constituting the domain. It is well known that the sign of the coefficient of the \(x^{2}\) term determines whether the graph of a quadratic function will have a "hill" or a "valley." On the basis of the present problem, which sign is associated with the hill? Supply an intuitive explanation for this.
Problem 4
Does any of the foltowing, drawn in a rectangular coordinate plane, represent a function? (a) A circle (b) A triangle (c) A rectangle (d) A downward-sloping straight line
Problem 4
Which of the following statements are valid? (a) \(A \cup A=A\) (b) \(A \cap A=A\) (c) \(A \cup \varnothing=A\) \((d) A \cup U=U\) \((e) A \cap \varnothing=\varnothing\) \((f) A \cap U=A\) (g) The complement of \(\tilde{A}\) is \(A\)
Problem 4
Graph the function \(y=36 / x,\) assuming that \(x\) and \(y\) can take positive values only. Next, suppose that both variables can take negative values as well; how must the graph be modified to reflect this change in assumption?
Problem 5
Condense the following expressions: (a) \(x^{4} \times x^{15}\) (b) \(x^{a} \times x^{b} \times x^{c}\) (c) \(x^{3} \times y^{3} \times z^{3}\)
Problem 5
If the domain of the function \(y=5+3 x\) is the set \(\\{x | 1 \leq x \leq 9\\}\), find the range of the function and express it as a set.
