1.1: Real Numbers, Functions and Graphs

This section describes real numbers as being either finite, infinite but not repeating and repeating. A finite number is a number such as 1/2 which when converted to decimal form is only 0.5. An infinite but not repeating number is a number such as pi, where the decimal places continue on to infinity, but the sequence is never repeated and a repeating number is one where a certain sequence of numbers is repeated over and over again. Numbers can also be defined as rational and irrational. A rational number will be finite or repeating, while an irrational number will be infinite but not repeating. Intervals are used to understand how numbers relate to one another. There are closed intervals, denoted by brackets [ ] which include the end numbers; open intervals, denoted by parentheses ( ) which exclude the end numbers and half open intervals, denoted by a bracket and a parenthesis [ ) or ( ] which either include or exclude the number on the end with the corresponding bracket or parenthesis.

Graphing is extremely helpful in calculus to help define the domain and range of equations. Producing a graph for a specific equation is helpful because it gives the person trying to solve the equation a clear image of what they are dealing with. A function is an equation that consists of inputs and outputs. Each input within a function can only have one specific output, making the individual inputs and outputs unique. Graphs of functions can be both shifted and scaled either vertically or horizontally depending on the way the equation is written.



Problems:

#29: Suppose that
x-4
is less than or equal to 1

a. What is the maximum possible value of
x+4
?

To find the answer to this problem I began to plug numbers in for ‘x.’ I used numbers starting with 1 and plugged them into the original equation to ensure that the answer would be less than or equal to 1. The largest possible number that I found which would work in the original equation was 5, with 5-4 = 1. This means that when plugged into the equation
x+4
, the maximum value for that equation would be 9.

b. Show that
x^2-16
is less than or equal to 9.

To solve this question I used the complete the square method. I broke up the equation into (x+4) and (x-4). I plugged in 5 for x, which is the largest possible value for x that would make this equation true. Using 5 for x, x^2 becomes 25 – 16 = 9. Any number plugged in for x that is less than or equal to 5 will make this equation true.

#48: Find the domain and range of the function g(t) = cos 1/t

To solve this problem, I used what I know about functions. I broke up the function into two parts: 1/t and cos x to help me figure the problem out. T could be any number other than zero because if t were equal to zero it would make the function undefined. The domain of the cosine function is from negative infinity to infinity which means that the domain for the entire function g(t)= cos 1/t should be from negative infinity to infinity. The range of the function 1/t is from (negative infinity, 0] and [0 to infinity). Zero is excluded because it would make the function undefined. The range of a cosine function is from (-1,1). When the two pieces of the function are combined I think that the range of the function g(t)= cos 1/t is from (-1, 0] and [0 to 1).