MATH 131-503 Fall 2015
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Chapter 1 Functions and Models
1.1 Four Ways to Represent a Function
• A function f is a rule that assigns to each element x in a set D exactly one element, called
f (x), in a set E.
• The set D is called the domain of the function.
The range of f is the set of all possible values of f (x) as x varies throughout the domain.
• The number f (x) is the value of f at x and is read “f of x”.
• A symbol that represents an arbitrary number in the domain of a function f is called an
independent variable.
• A symbol that represents a number in the range of f is called a dependent variable.
Examples:
1. Arrow diagram: which of the followings are functions?
2. The area A of a circle depends on the radius r of the circle. The rule that connects r and
A is given by the equation A = πr2 . With each positive number r there is associated one value of
A, and we say that A is a function of r. r is the independent variable and A is the dependent variable.
There are four possible ways to represent a function:
• verbally (by a description in words)
• numerically (by a table of values)
• visually (by a graph)
• algebraically (by an explicit formula)
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MATH 131-503 Fall 2015
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The most common method for visualizing a function is its graph.
• If f is a function with domain D, then its graph is the set of ordered pairs {(x, f (x)) | x ∈ D}.
• A function f is called increasing on an interval I if f (x1 ) < f (x2 ) whenever x1 < x2 in I.
• It is called decreasing on I if f (x1 ) > f (x2 ) whenever x1 < x2 in I.
Examples:
3. (p. 21) The graphs of f and g are given.
(a) f (−4)=
g(3) =
(b) For what values of x is f (x) = g(x)?
(c) Estimate the solution of the equation
f (x) = −1.
(d) On what interval is f decreasing? Decreasing?
(e) State the domain and range of g.
4. (p. 14) Sketch the graph and find the domain and range of each function.
(a) f (x) = 2x − 1
(b) g(x) = x2
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MATH 131-503 Fall 2015
5. Find the domain of f (x) = √
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1−x
x2 − 8x
6. (p. 18) Find f (−1), f (2) and the domain of the piecewise defined function
(
1 − x, if x ≤ 1
f (x) =
x2 ,
if x > 1
and sketch the graph.
7. If f (x) = 4x2 − x + 5, find the following.
(a) f (1)
(b) f (−a)
(c) (f (a))2
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MATH 131-503 Fall 2015
(d)
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f (2 + h) − f (2)
h
The Vertical Line Test: A curve in the xy-plane is the graph of a function of x if and only if no
vertical line intersects the curve more than once.
Examples. Determine whether each graph represents a function of x.
8. g(x) is the function below.
9. f (x) is the function below.
Definitions:
• The absolute value of a number x is
|x| = x
if x ≥ 0
|x| = −x if x < 0
• If a function f satisfies f (−x) = f (x) for every number x in its domain, then f is called an
even function. The graph of an odd function is symmetric about the y-axis.
• If a function f satisfies f (−x) = −f (x) for every number x in its domain, then f is called an
odd function. The graph of an odd function is symmetric about the origin.
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Examples:
10. (p. 20) Determine whether h(x) = 2x − x2 and g(t) = x5 + x are even, odd, or neither even nor
odd.
11. (p. 24) A box with an open top is to be constructed from a rectangular piece of cardboard with
dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding
up the sides as in the figure. Express the volume V of the box as a function of x.
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12. (p. 24) A function f has domain [−5, 5] and a portion of its graph is shown.
(a) Complete the graph of f if f is even.
(b) Complete the graph of f if f is odd.
13. (a) If the point (5, 7) is on the graph of an even function, what other point must also be on the
graph?
(b) If the point (5, 7) is on the graph of an odd function, what other point must also be on the graph?
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