MATH 131-505 Spring 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. 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.
2. arrow diagram
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)
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.
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MATH 131-505 Spring 2015
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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?
(e) State the domain and range of g.
4. (p. 14) Sketch the graph and find the domain and range of each function.
(b). g(x) = x2 .
(a). f (x) = 2x − 1.
5. Find the domain of f (x) = √
1−x
x2 − 8x
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MATH 131-505 Spring 2015
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c
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6. (p. 18) Find f (−1), f (1) and the domain of the piecewise defined function
(
1 − x, if x ≤ 1
f (x) =
x2 ,
if x > 1
and ketch the graph.
7. If f (x) = 4x2 − x + 5, find the following.
(a). f (1)
(b). f (−a)
(c).
f (2 + h) − f (2)
h
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MATH 131-505 Spring 2015
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Liu
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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.
• If a function f satisfies f (−x) = −f (x) for every number x in its domain, then f is called an
odd function.
Examples:
10. (p. 20) Determine whether h(x) = 2x − x2 and g(t) = x5 + x are even, odd, or neither even nor
odd.
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MATH 131-505 Spring 2015
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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.
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.
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