Chapter 3

Evaluate each sum. $$\sum_{i=1}^{6} i$$

Solve the following equations. $$\left[\begin{array}{ccc}{x-1} & {2} & {0} \\ {0} & {y+3} & {4} \\ {-3} & {1} & {z+2}\end{array}\right]=\left[\begin{array}{rrr}{-2} & {2} & {0} \\ {0} & {-1} & {4} \\ {-3} & {1} & {-2}\end{array}\right]$$

Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=2 x-1\) and \(g(x)=x^{2}+1 .\) Find: $$(g \circ f)(2)$$

Show that in any 11 -digit integer, at least two digits are the same.

Evaluate each, where \(n\) is an integer. $$\lfloor n+1 / 2\rfloor$$

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). Express \(-40^{\circ} \mathrm{C}\) on the Fahrenheit scale.

Solve the following equations. $$\left[\begin{array}{ccc} x-1 & 2 & 0 \\ 0 & y+3 & 4 \\ -3 & 1 & z+2 \end{array}\right]=\left[\begin{array}{rrr} -2 & 2 & 0 \\ 0 & -1 & 4 \\ -3 & 1 & -2 \end{array}\right]$$

Determine if each function is the identity function. $$\begin{array}{c|cccc} x & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} \\ \hline f(x) & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} \end{array}$$

Evaluate each sum. $$\sum_{k=0}^{4}(3+k)$$

Solve the following equations. $$\left[\begin{array}{ccc}{x-y} & {-1} & {0} \\ {-3} & {y-z} & {2} \\ {4} & {-5} & {z-x}\end{array}\right]=\left[\begin{array}{rrr}{3} & {-1} & {0} \\\ {-3} & {-4} & {2} \\ {4} & {-5} & {1}\end{array}\right]$$

Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=2 x-1\) and \(g(x)=x^{2}+1 .\) Find: $$(f \circ g)(-1)$$

Show that in any 27 -letter word, at least two letters are the same.

The Celsius and Fahrenheit scales are related by the formula \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\) $$ g(x)=\left\\{\begin{array}{ll}{2|x|+3} & {\text { if } x \leq 0} \\ {5} & {\text { if } 0 < x \leq 3 \text { . Compute each. }} \\ {-x^{2}} & {\text { otherwise }}\end{array}\right. $$

Evaluate each, where \(n\) is an integer. $$\lfloor n / 2\rfloor$$

Solve the following equations. $$\left[\begin{array}{ccc} x-y & -1 & 0 \\ -3 & y-z & 2 \\ 4 & -5 & z-x \end{array}\right]=\left[\begin{array}{rrr} 3 & -1 & 0 \\ -3 & -4 & 2 \\ 4 & -5 & 1 \end{array}\right]$$

Evaluate each sum. $$\sum_{j=0}^{4}(j-1)$$

Find the additive inverse of each matrix. $$\left[\begin{array}{rr} 2 & -3 \\ 0 & 4 \end{array}\right]$$

Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=2 x-1\) and \(g(x)=x^{2}+1 .\) Find: $$(g \circ f)(x)$$

Six positive integers are selected. Show that at least two of them will have the same remainder when divided by five.

Evaluate each, where \(n\) is an integer. $$[n+1 / 2]$$

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). $$g(-3.4)$$

Evaluate each, where \(n\) is an integer. $$\lceil n+1 / 2\rceil$$

Evaluate each sum. $$\sum_{i=-1}^{4} 3$$

Find the additive inverse of each matrix. $$\left[\begin{array}{rrr} 1 & -2 & 3 \\ 3 & 3 & -1 \end{array}\right]$$

A \(C++\) identifier contains 37 alphanumeric characters. Show that at least two characters are the same.

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$f(x)=|x|, x \in \mathbb{R}$$

Evaluate each, where \(n\) is an integer. $$\lceil n / 2\rceil$$

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). $$g(0)$$

Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=2 x-1\) and \(g(x)=x^{2}+1 .\) Find: $$(f \circ g)(x)$$

\(\mathrm{A} \mathrm{C}++\) identifier contains 37 alphanumeric characters. Show that at least two characters are the same.

Evaluate each sum. $$\sum_{n=0}^{4}(3 n-2)$$

Find the additive inverse of each matrix. $$\left[\begin{array}{rrr} 0 & -3 & -2 \\ 1 & -2 & 4 \\ 2 & -5 & 6 \end{array}\right]$$

Let \(f(x)=\lfloor x\rfloor\) and \( g(x)=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(g \circ f)(-2.3)$$

Show that in any group of eight people, at least two must have been born on the same day of the week.

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$g(x)=2^{x}, x \in \mathbb{R}$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x+y\rfloor$$

Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(g \circ f)(-2.3)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x+y\rfloor$$

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). $$g(0.27)$$

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$A-B$$

Evaluate each sum. $$\sum_{j=-2}^{2} j(j-2)$$

Let \(f(x)=\lfloor x\rfloor\) and \( g(x)=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-2.3)$$

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$h(x)=\lg x, x \in \mathbb{R}^{+}$$

Show that in any group of 13 people, at least two must have been born in the same month.

The Celsius and Fahrenheit scales are related by the formula \(F=\frac{9}{5} C+32\). $$g(4.5)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x\rfloor+\lfloor y\rfloor$$

Let \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 5 \\ 0 & 0 & 1\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 0 & 1 & 2\end{array}\right] .\) Find each. $$A-B$$

Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-2.3)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x\rfloor+\lfloor y\rfloor$$

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$B+C$$