Practice Problems 2

Lecture 2: Relations and Functions; Summation Notation; Necessary and Sufficient Conditions

Problem numbers refer to the textbook (Chiang & Wainwright, 4th edition). These are not graded, but you should attempt them all before the quiz. Each problem has a Show solution toggle: try it yourself first, then expand to check.

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Exercise 2.4

5. If the domain of the function \(y=5+3x\) is the set \(\{x \mid 1 \leq x \leq 9\}\), find the range of the function and express it as a set.

Show solution

For this function, when \(x=1\), \(y=8\), and when \(x=9\), \(y=32\). So the range is \[ f(X) =\{y \mid 8 \leq y \leq 32\} \]

Note: it is not always the case that extreme values of the domain correspond to extreme values of the range. For example, consider \(y=x^{2}\) with domain \(\{x \mid -2 \leq x \leq 2\}\); the range here is \(\{y \mid 0 \leq y \leq 4\}\).

7. In the theory of the firm, economists consider the total cost \(C\) to be a function of the output level \(Q\): \(C=f(Q)\).

  1. According to the definition of a function, should each cost figure be associated with a unique output level?
  2. Should each level of output determine a unique cost figure?
Show solution
  1. No
  2. Yes

8. If an output level \(Q_1\) can be produced at a cost of \(C_1\), then it must also be possible (by being less efficient) to produce \(Q_1\) at a cost of \(C_1+\$1\), or \(C_1+\$2\), and so on. Thus it would seem that output \(Q\) does not uniquely determine total cost \(C\). If so, to write \(C=f(Q)\) would violate the definition of a function. How, in spite of this reasoning, would you justify the use of the function \(C=f(Q)\)?

Show solution

For each output level, we would want to produce at the lowest cost. The cost function \(C=f(Q)\) records that minimum cost, which is unique.

Exercise 2.5

1. Graph the following functions and find their inverse functions.

  1. \(y = 16 + 2x\)
  2. \(y = 8-2x\)
  3. \(y = 2x+12\)
Show solution

\(y = 16 + 2x, \quad f^{-1}(y) = \dfrac{y-16}{2}\)

Graph of the straight line y equals 16 plus 2x: an upward-sloping line with slope 2 crossing the y-axis at 16

\(y = 8-2x, \quad f^{-1}(y) = \dfrac{8-y}{2}\)

Graph of the straight line y equals 8 minus 2x: a downward-sloping line with slope negative 2 crossing the y-axis at 8

\(y = 2x+12, \quad f^{-1}(y) = \dfrac{y-12}{2}\)

Graph of the straight line y equals 2x plus 12: an upward-sloping line with slope 2 crossing the y-axis at 12

Exercise 4.2

6. Expand the following summation expressions:

  1. \(\sum_{i=2}^5 x_i\)
  2. \(\sum_{i=5}^8 a_i x_i\)
  3. \(\sum_{i=1}^4 b x_i\)
  4. \(\sum_{i=1}^n a_i x^{i-1}\)
  5. \(\sum_{i=0}^3(x+i)^2\)
Show solution
  1. \(x_{2}+x_{3}+x_{4}+x_{5}\)
  2. \(a_{5} x_{5}+a_{6} x_{6}+a_{7} x_{7}+a_{8} x_{8}\)
  3. \(b x_{1}+b x_{2}+b x_{3}+b x_{4}\)
  4. \(a_{1}+a_{2} x+a_{3} x^{2}+\ldots+a_{n} x^{n-1}\)
  5. \(x^{2}+(x+1)^{2}+(x+2)^{2}+(x+3)^{2}\)

8. Show that the following are true:

  1. \(\left(\sum_{i=0}^n x_i\right)+x_{n+1}=\sum_{i=0}^{n+1} x_i\)
  2. \(\sum_{j=1}^n a b_j y_j=a \sum_{j=1}^n b_j y_j\)
  3. \(\sum_{j=1}^n\left(x_j+y_j\right)=\sum_{j=1}^n x_j+\sum_{j=1}^n y_j\)
Show solution

(a) \[ \left(\sum_{i=0}^{n} x_{i} \right)+x_{n+1} = x_{0}+x_{1}+x_{2}+\ldots+x_{n+1} = \sum_{i=0}^{n+1} x_{i} \]

(b) \[ \begin{aligned} \sum_{j=1}^{n} a b_{j} y_{j} &= a b_{1} y_{1}+a b_{2} y_{2}+\ldots+a b_{n} y_{n} \\ &= a\left(b_{1} y_{1}+b_{2} y_{2}+\ldots+b_{n} y_{n}\right) \\ &= a \sum_{j=1}^{n} b_{j} y_{j} \end{aligned} \]

(c) \[ \begin{aligned} \sum_{j=1}^{n}\left(x_{j}+y_{j}\right) &= \left(x_{1}+y_{1}\right)+\left(x_{2}+y_{2}\right)+\ldots+\left(x_{n}+y_{n}\right) \\ &= x_{1}+x_{2}+\ldots+x_{n}+y_{1}+y_{2}+\ldots+y_{n} \\ &= \sum_{j=1}^{n} x_{j}+\sum_{j=1}^{n} y_{j} \end{aligned} \]

Exercise 5.1

1. In the following paired statements, let \(p\) be the first statement and \(q\) the second. Which is true for each case: \(p \Rightarrow q\), \(p \Leftarrow q\), or \(p \Leftrightarrow q\)?

  1. It is a holiday; it is Thanksgiving Day.
  2. A geometric figure has four sides; it is a rectangle.
  3. Two ordered pairs \((a, b)\) and \((b, a)\) are equal; \(a\) is equal to \(b\).
  4. A number is rational; it can be expressed as a ratio of two integers.
  5. A \(4 \times 4\) matrix is nonsingular; the rank of the \(4 \times 4\) matrix is 4. (skip for now)
  6. The gasoline tank in my car is empty; I cannot start my car.
  7. The letter is returned to the sender with the marking “addressee unknown”; the sender wrote the wrong address on the envelope.
Show solution
  1. \(q \implies p\)
  2. \(q \implies p\)
  3. \(q \iff p\)
  4. \(q \iff p\)
  5. \(q \iff p\)
  6. \(p \implies q\)
  7. \(q \implies p\)