Chapter 1: Introduction to Functions and Graphs

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Chapter 1 Introduction to Functions and Graphs

• Section 1.4 Types of Functions and Their Rates
  of Change

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Several Common Types of Functions

• While the definition of a function in mathematics
  is very general, there are several more specific
  types of functions which are very useful in
  modeling applications.
• The three basic types of functions we will study
  in this section are constant, linear, and
  nonlinear functions.
• Constant FunctionA function f represented by
  ,where b is a fixed real number, is
  a constant function.
• Example 1 Let . Then f is a
  constant function.
• Notice that the graph of a constant (continuous)
  function is a perfectly horizontal line.
• Linear FunctionA function f represented by
  , where a and b are fixed real
  numbers (constants), is a linear function.

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Several Common Types of Functions (contd)

• Example 2 Let . Then
  f and g are both linear functions.
• Notice that the graph of a linear function is
  simply a straight line.
• Notice also that every constant function is also
  a linear function (since a horizontal line is, in
  fact, a line)but that there are linear functions
  (such as f and g above) which are not constant
  functions. Thus, when asked to completely
  classify a given function as constant, linear, or
  nonlinear, any constant function should be
  classified as both constant and linear, while
  every other linear function must simply be
  classified as linear.
• Note that we can determine whether a function is
  constant, linear, or nonlinear, by its graph. As
  stated, the graph of a constant function is a
  perfectly horizontal line. The graph of a linear
  function is a slanted (sloped) line. The graph of
  a nonlinear function is a curve.

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Several Common Types of Functions (contd)

• Note that we can also determine whether a
  function is constant, linear, or nonlinear
  without having to see its graph. We can look
  simply at a functions symbolic representation.
• Determining Constant, Linear, Nonlinear Functions
• A function for which the independent variable
  takes on only a power (exponent) of 0
  (zero)typically, not written explicitlyis a
  constant function.
• A function for which the independent variable
  takes on only a power of 1 (one) is a linear
  function.
• A function whose independent variable takes on
  any other power than zero or one is a nonlinear
  function. (For our current purposes, we shall
  take this to be the definition of a nonlinear
  function).
• Example 3 Let
  . Then f is a constant function (x is
  raised to the zero power), g is a linear function
  (x is raised to the 1st power), and h is a
  nonlinear function (x is raised to a power other
  than zero or onespecifically, power 2). Note
  that since f is a constant function, it is also
  a linear function.

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Function Characteristics Rates of Change

• Having these important types of functions, we
  proceed to study important characteristics
  concerning our functions. One of the most
  important things we often need to know is how
  much a functions output value (y-value) changes
  as we move from a particular x-value to another
  particular x-value. (This notion of a functions
  rate of change is, in fact, the conceptual
  foundation for the differential calculus). We
  proceed to define the average rate of change of a
  function from points x1 to x2 in the domain of
  that function.
• Average Rate of Change
• Let be distinct points on
  the graph of a function f.The average rate of
  change of f from x1 to x2 is

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Function Characteristics Rates of Change (contd)

• Example 4 Let . What is the
  average rate of change of f on the interval
  2,5? (That is, from x2 to x5).
• Solution We let x1 2, x2 5. We have y1
  f(x1) f(2) 2(4) 8, and y2 f(x2) f(5)
  2(25) 50. Thus, the average rate of change of f
  on the interval 2,5 is given by (50-8)/(5-2)
  (42)/(3) 14.
• Problem What is the average rate of change of
  the function on the interval
  x1,x2?

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Function Characteristics Slope

• Notice that for the given f below, it doesnt
  matter which values we assign to x1 and x2 the
  average rate of change of f remains the same no
  matter which interval of its domain we consider.
  That is, this particular function exhibits a
  constant rate of change. This is because f is a
  linear function.
• The average rate of change of a linear function
  is called the slope of that linear function
  (i.e., the slope of the line represented by that
  function) and is conventionally denoted by the
  letter m.
• Problem Let . Then f is a
  linear function. Find the slope of the line
  represented by f.
• Since any two points determine a unique line in
  the cartesian plane, we can find the slope of the
  line passing through any two points (x1,y1) and
  (x2,y2).
• Problem Find the slope of the line passing
  through the points (4,6) and (2,5).
• Note that it doesnt matter which point we choose
  as our first, and which as our second.

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A Few Comments About Slope

• Notice that the slope of any constant function
  (i.e., horizontal line) is equal to zero. (Why?)
• The slope of a line can be thought of in the
  following manner if a line f has slope m, then
  for every one unit we move right on the x-axis,
  we must move m units up or down (depending on
  whether m is positive or negative) to find
  another point on the line. (Example?)
• Suppose that a line f has slope m. If mgt0, then
  the graph of f rises as we move from left to
  right on the x-axis. This line is said to have
  positive slope. If mlt0, then the graph of f
  falls as we move from left to right on the
  x-axis. This line is said to have negative slope.
  (Examples?) If the line f has slope m0, then
  of course f is a constant function and so the
  line is horizontal.
• A perfectly vertical line is said to have
  undefined slope (or, sometimes, infinite slope).
  (Why is this slope undefined?) Note that, in
  any case, undefined slope is not a problem since
  slope is defined for functions. A vertical line
  is not a function. (Why?)
• Finally, note that for a nonlinear function, the
  average rate of change of that function on an
  interval a,b is really only the slope of the
  line passing through the points (a, f(a)) and (b,
  f(b)).
• Homework for this section