Practice Problems 2

Lecture 2: Functions, Summation Notation, and Logical 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 Solution toggle, and a few have one per part: try it yourself first, then expand to check.

Exercise 2.4

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

Solution

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

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 $\{x\mid -2\le x\le 2\}$; the range here is $\{y\mid 0\le y\le 4\}$.

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?

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 $C_1+\mathrm{\$}1$, or $C_1+\mathrm{\$}2$, 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 $C=f(Q)$?

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. $y=16+2x$

Solution

$f^{-1}(y)={\displaystyle \frac{y-16}{2}}$

2. $y=8-2x$

Solution

$f^{-1}(y)={\displaystyle \frac{8-y}{2}}$

3. $y=2x+12$

Solution

$f^{-1}(y)={\displaystyle \frac{y-12}{2}}$

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. $\sum _{i=0}^3(x+i{)}^2$

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. $a_1+a_{2}x+a_3{x}^2+\dots +a_n{x}^{n-1}$
5. $x^2+(x+1{)}^2+(x+2{)}^2+(x+3{)}^2$

8. Show that the following are true:

1. $\left(\sum _{i=0}^n{x}_i\right)+x_{n+1}=\sum _{i=0}^{n+1}{x}_i$

Solution

$$\left(\sum _{i=0}^n{x}_i\right)+x_{n+1}=x_0+x_1+x_2+\dots +x_{n+1}=\sum _{i=0}^{n+1}{x}_i$$

2. $\sum _{j=1}^{n}a{b}_j{y}_j=a\sum _{j=1}^n{b}_j{y}_j$

Solution

$$\begin{array}{rl}\sum _{j=1}^{n}a{b}_j{y}_j& =a{b}_1{y}_1+a{b}_2{y}_2+\dots +a{b}_n{y}_n\\ & =a(b_1{y}_1+b_2{y}_2+\dots +b_n{y}_n)\\ & =a\sum _{j=1}^n{b}_j{y}_j\end{array}$$

3. $\sum _{j=1}^n(x_j+y_j)=\sum _{j=1}^n{x}_j+\sum _{j=1}^n{y}_j$

Solution

$$\begin{array}{rl}\sum _{j=1}^n(x_j+y_j)& =(x_1+y_1)+(x_2+y_2)+\dots +(x_n+y_n)\\ & =x_1+x_2+\dots +x_n+y_1+y_2+\dots +y_n\\ & =\sum _{j=1}^n{x}_j+\sum _{j=1}^n{y}_j\end{array}$$

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 $p\iff q$?

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. The gasoline tank in my car is empty; I cannot start my car.
6. The letter is returned to the sender with the marking “addressee unknown”; the sender wrote the wrong address on the envelope.

Solution

1. $q⟹p$
2. $q⟹p$
3. $q⟺p$
4. $q⟺p$
5. $p⟹q$
6. $q⟹p$