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Section 1.5
The graph of a function f is the collection of ordered pairs (x, f (x)) where x is in the domain of f.
(2, –2) is on the graph of f (x) = (x – 1)2 – 3.
x
y
4
-4(2, –2)
f (2) = (2 – 1)2 – 3 = 12 – 3 = – 2
f (x) = (x – 1)2 – 3
x
y
4
-4
The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.
Domain
Range
Example 1
Find the domain and range of the function
f (x) = from its graph.3x
x
y
– 1
1
The domain is [–3,∞).
The range is [0,∞).
Range
Domain
(–3, 0)
3f x x
x
y
4
-4
Vertical Line Test
A relation is a function if no vertical line intersects its graph in more than one point.
This graph does not pass the vertical line test. It is not a function.
This graph passes the vertical line test. It is a function.
y = x – 1x = | y – 2|
x
y
4
-4
A function f is:
• increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2),
• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),
• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).
• The x-value that a graph changes direction is not in the interval.
The graph of y = f (x):
• decreases on (– ∞, –1),• constant on (–1, 1),
• increases on (1, ∞).
-4 -2 2 4
2
4
6
x
y
(-1, 2) (1, 2)
A function value f (a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies
f (a) ≤ f (x).
A function value f (a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies
f (a) ≥ f (x).
x
y
Relative minimum
Relative maximum
Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1.
1 1Relative Minimum: , 1
3 3
A function f is even if for each x in the domain of f, f (–x) = f (x).
x
yf (x) = x2
f (–x) = (–x)2 = x2 = f (x)
f (x) = x2 is an even function.
Symmetric with respect to the y-axis.
A function f is odd if for each x in the domain of f, f (– x) = – f (x).
x
y
f (x) = x3
f (– x) = (– x)3 = –x3 = – f (x)
f (x) = x3 is an odd function.Symmetric with respect to the origin.
Example 2
Determine whether the function is even, odd, or neither. Then describe the symmetry.
a. f (x) = x6 – 2x2 + 3
b. g(x) = x3 − 5x
a. f (x) = x6 – 2x2 + 3
f (-x) = (-x)6 – 2(-x)2 + 3
= x6 – 2x2 + 3
f (x) is an even function and has symmetry with the y-axis.
b. g(x) = x3 − 5x
g(-x) = (-x)3 – 5(-x)
= −x3 + 5x
= −(x3 – 5x)
= −g(x)
g(x) is an odd function and has symmetry with the origin.
Zeros of a Function
If the graph of a function of x has an x-intercept (a, 0), then a is a zero of the function.
Definition:
The zeros of a function f of x are the x-values for which f (x) = 0.
Example 3
Find the zeros of the function
f (x) = 2x2 + 13x – 24.
Solve by factoring.
2x2 + 13x – 24 = 0
(2x + 3)(x – 8) = 0
2x + 3 = 0 or x – 8 = 03
2x 8x
Average Rate of Change
For a linear graph the rate of change is constant between points and is called the slope of the line.
For a nonlinear graph whose slope changes at each point, the average rate of change between any two points (x1, f (x1)) and
(x2, f (x2)) is the slope on the line through the two points.
The line through the two points is called the secant line, and the slope of this line is denoted as msec.
Average rate of change
of f from x1 to x2
2 1
2 1
f x f x
x x
change in
change in
y
x
secm
x1 x2
(x2, f (x2))
(x1, f (x1))Secant Line
Example 4
Find the average rates of change of
f (x) = x2 – 2x if x1 = -2 and x2 = 1.
2 1
2 1
f x f x
x x
1 2
1 2
f f
1 8
3
3
Example 5
Write the height h of the rectangle as a function of x.
The rectangle is bound
by two graphs.
f (x) = -x2 + 4x – 1 and
g(x) = 2.
h = top – bottom
h = f (x) – g(x)
2 4 1 2h x x 2 4 3x x
Example 6
Write the length L of the rectangle as a function of y.
The rectangle is bound
by two graphs.
x = ½y2 and
x = 0.
L = right – left
210
2L y
21
2y
