Math 150 – Fall 2015
Section 4B
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Section 4B – Graphs of Functions
Definition. If f is a function with domain A, the graph of f is the set of all ordered
pairs (x, f (x)), where x ∈ A. That is, y = f (x).
Example 1. Use the graph of function f below to evaluate each of the following.
f (−1) =
f (2) =
f (−3) =
f (1) =
Example 2. Graph the function f (x) =
√
x − 4.
Find the y-intercept of a function: For any function, we can calculate the yintercept by finding the output value when x = 0, or f (0).
Find the x-intercept of a function: We can calculate the x-intercept by finding the
input when the output is 0. That is, set f (x) = 0 and solve for x.
Example 3. Find the x- and y-intercepts of the graph of f (x) = x3 − 9x.
Vertical Line Test
Theorem. A set of points in the coordinate plane represents a function if and only if
no vertical line intersects the graph in more than one point.
Example 4. Determine whether each of the following graphs represents a function.
Math 150 – Fall 2015
Section 4B
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Domain and Range from Graphs
We can find the domain and range of a function by reading its graph. The domain is a
projection of the graph onto the x-axis, while the range is the projection of the graph
onto the y-axis as shown below.
Example 5. Find the domain and range of f (x) =
√
x − 4.
Example 6. Use the following graphs to find the domain and range of the functions.
Math 150 – Fall 2015
Section 4B
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Catalog of Functions
1. Constant Function: f (x) = c
y-intercept: (0, c)
Domain: (−∞, ∞)
x-intercept: none
Range: {c}
2. Identity Function: f (x) = x
y-intercept: (0, 0)
Domain: (−∞, ∞)
x-intercept: (0, 0)
Range: (−∞, ∞)
3. Squaring Function: f (x) = x2
y-intercept: (0, 0)
Domain: (−∞, ∞)
x-intercept: (0, 0)
Range: [0, ∞)
4. Cubing Function: f (x) = x3
y-intercept: (0, 0)
Domain: (−∞, ∞)
x-intercept: (0, 0)
Range: (−∞, ∞)
5. Square Root Function: f (x) =
√
x
y-intercept: (0, 0)
Domain: [0, ∞)
x-intercept: (0, 0)
Range: [0, ∞)
6. Absolute Value Function: f (x) = |x|
y-intercept: (0, 0)
Domain: (−∞, ∞)
x-intercept: (0, 0)
Range: [0, ∞)
Math 150 – Fall 2015
Section 4B
7. Reciprocal Function: f (x) =
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1
x
y-intercept: none
Domain: (−∞, 0) ∪ (0, ∞)
x-intercept: none
Range: (−∞, 0) ∪ (0, ∞)
Piecewise Functions
Example 7. Graph the following piecewise functions.
2x
√ + 3 if x < 0
(a) f (x) =
x
if x ≥ 0
(b) f (x) =
− 12 x +
|x|
1
2
x < −1
x ≥ −1
 3
x < −1
 x + 1,
3,
x = −1
(c) f (x) =
 2
x ,
−1 < x < 2
Example 8. For the previous graph in (c), find the following:
(a) Domain:
(b) Range:
(c) x-intercept:
(d) y-intercept:
(e) f (0) =
(f) f (−2) =
(g) f (−1) =
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Section 4B
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Increasing, Decreasing, and Constant
Definition. A function f is increasing on an interval I if and only if for every x1 <
x2 ∈ I, f (x1 ) < f (x2 ).
Figure 1: Graph goes uphill from left to right. As x increases the values of f (x) increase.
Definition. A function f is decreasing on an interval I if and only if for every
x1 < x2 ∈ I, f (x1 ) > f (x2 ).
Figure 2: Graph goes downhill from left to right. As x increases the values of f (x)
decrease.
Definition. A function f is constant on an interval I if and only if for every x1 , x2 ∈ I,
f (x1 ) = f (x2 ).
Figure 3: Graph is horizontal over I.
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Section 4B
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Example 9. Determine where the functions are increasing, decreasing, and/or constant.
Example 10. Use the graph to find the following for each function:
(1) Domain, (2) Range, (3) Intercepts, and
(4) Intervals where f is increasing, decreasing, and constant
Example 11. Graph a function f (x) that has domain (−2, ∞), range [−∞, 5], is
increaing on (−2, 0), constant on (0, 2), and decreasing on (2, 0).