1
c
Kathryn Bollinger, August 31, 2010
1.1 - Four Ways to Represent a Function Functions arise when one quantity depends on another. Def: A function f is a rule that assigns to each element in a set D exactly one element, called f (x), in a set R. • Set D is called the domain of the function. =⇒ It is the set of all possible input values, x; the set of
variables
• Set R is called the range of the function. =⇒ It is the set of all possible output values, f (x); the set of =⇒ f (x) is read “f of x”
variables
Ex: Find the domain and range of each correspondence below and then determine whether each is a function of x. If it is a function of x, then find the value, f (−1). y
x
y
−3
−2
−1
0
1
1
6
5
10
7
2x + 4y = 8
7
x
7
Ex: Given the graph of f to the right, find the following: y
(a) f (5)
7
(b) Where f (x) = 1 (c) Domain of f
x
7
(d) Range of f
2
c
Kathryn Bollinger, August 31, 2010
Vertical Line Test: A curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once. Ex: Find the domain and range of each correspondence below and then determine whether each is a function of x. y
y
6
2 x
2
4
6
1 0 0 1 0 1
6 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 2 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 2 4 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0 1 0000000000000 1111111111111 0 1
y
6
2 x
6
x
2
4
6
Finding the Domain of a Function Algebraically Find all of the values of the independent variable (x) that produce real values for the dependent variable (y). Ex: Find the domain of the following functions. Write your answer using interval notation. (a) g(x) = x8 + 10x
(b) f (x) =
x2 − 4 x+2
(c) k(x) =
√
(d) r(x) =
(e) h(x) =
7−x
√
x+3 x−1
√ 3
x+4
3
c
Kathryn Bollinger, August 31, 2010
Ex: If f (x) = 3x2 + 2x + 1, evaluate the difference quotient,
f (a + h) − f (a) for h 6= 0 h
Ex: If f (x) =
√
Ex: If f (x) =
1 f (x) − f (a) , evaluate the difference quotient, for x 6= a x x−a
x, evaluate the difference quotient,
f (x + h) − f (x) for h 6= 0 h
4
c
Kathryn Bollinger, August 31, 2010
Four Ways to Represent a Function 1. Verbally (by a description in words) 2. Numerically (by a table of values) 3. Visually (by a graph) 4. Algebraically (by an explicit formula)
Many times, functions are described more “naturally” by one method than another. For example, * Rather than looking at a table of values for the population of a country based on the year, it is easier to look at a graph to quickly see the trend. * It is more useful to represent the area of a circle as a function of its radius algebraically (A(r) = πr 2 ), than it is to compile a table of values.
Ex: (# 18) You place a frozen pie in an oven and bake it for an hour. Then, you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then, sketch a rough graph of the temperature of the pie as a function of time.
Ex: (#10) The graph below shows the height of the water in a bathtub as a function of time. Give a verbal description of what you think happened. height (inches)
25 15 5 time (min) 5
15
5
c
Kathryn Bollinger, August 31, 2010
Ex: Temperature readings T (in ◦ F ) were recorded every two hours from 12:53 AM to 12:53 PM in College Station, TX on August 30, 2010. The time t was measured in hours from 12:53 AM. t T
0 82
2 80.1
4 79
6 79
8 84
10 79
12 88
(a) Use the readings to sketch a rough graph of T as a function of t.
(b) Use your graph to estimate the temperature at 11:53 AM.
Ex: Express the surface area of a cube, (a) As a function of the length of one side of the cube.
(b) As a function of the cube’s volume.
c
Kathryn Bollinger, August 31, 2010
6
Piecewise-Defined Functions: Functions whose definitions involve more than one rule. To graph, graph each rule over the appropriate portion of the domain.
Ex: For f (x) =
(
2x , x < −1 −4 + x2 , x ≥ −1
(a) Find f (−3), f (−1), and f (0).
(b) Make an accurate graph of f (x).
Ex: A taxi cab company in a certain town charges all customers a base fee of $5.00 per ride. They then charge an additional 50 cents/mile for the first 10 miles traveled and $1/mile for each mile over 10 miles. Write a piecewise function, C(x), for the cost of a cab ride if x represents the number of miles traveled.
7
c
Kathryn Bollinger, August 31, 2010
Def: The absolute value of a number, |a|, is the distance from a to 0 on the real number line. Since distances are non-negative, |a| ≥ 0. Thus, the absolute value function f (x) = |x| is defined by f (x) =
(
−x , x < 0 x ,x≥0
Increasing and Decreasing Functions A function f is called increasing on an interval I if f (x1 ) < f (x2 ) whenever x1 < x2 in I. A function f is called decreasing on an interval I if f (x1 ) > f (x2 ) whenever x1 < x2 in I. Ex: Given the graph of a function to the right: Over what interval(s) is the function increasing?
y
7
x
7
Over what interval(s) is the function decreasing?
Ex: Over what interval(s) is f (x) = |x| increasing? Decreasing?
8
c
Kathryn Bollinger, August 31, 2010
Symmetry • Even Functions ◦ f (−x) = f (x) for every number x in the function f ’s domain. ◦ The graph is symmetric with respect to the y-axis
Ex: Show f (x) =
x4 is even. x2 + 3
Ex: A function g has domain (−∞, ∞) and a portion of its graph is shown below. Complete the graph if is is known that g(x) is an even function
• Odd Functions ◦ f (−x) = −f (x) for every number x in the function f ’s domain. ◦ The graph is symmetric about the origin.
Ex: Show f (x) =
x3 is odd. x2 + 2
Ex: A function g has domain (−∞, ∞) and a portion of its graph is shown below. Complete the graph if is is known that g(x) is an odd function
