2. Nonlinear Functions, Conic Sections, and Nonlinear Systems
Chapter 12
Nonlinear Functions, Conic Sections,
and Nonlinear Systems

3. Additional Graphs of Functions; Composition
12.1
Additional Graphs of Functions;
Composition

4. 12.1 Additional Graphs of Functions; Composition
Objectives
Recognize the graphs of the elementary functions defined
by | x |, , and , and graph their translations.
Recognize and graph step functions.
Find the composition of functions. 1 x x
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5. 12.1 Additional Graphs of Functions; Composition
The Absolute Value Function
The domain of the absolute value function is (– ∞, ∞) and its range is [0, ∞). x y
x y +
(0, 0)
f ( x ) = | x |
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6. 12.1 Additional Graphs of Functions; Composition
The Reciprocal Function
For the reciprocal function, the domain and the range are both (– ∞, 0) U (0, ∞).
Notice the vertical asymptote at x = 0 and the
horizontal asymptote at y = 0 . x y
x y 1 3 2
x y 1 3 2 –
f ( x ) = 1 x
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7. 12.1 Additional Graphs of Functions; Composition
The Square Root Function
The domain of the square root function is [0, ∞). Because the principal
square root is always nonnegative, the range is
also [0, ∞). x y
x y 1 2 4
(0, 0)
f ( x ) = x
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8. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 1
Applying a Horizontal Shift
Graph f ( x ) = | x – 3 |.
This graph is found by shifting f ( x ) = | x | three units to the right.
The domain of this function is (– ∞, ∞) and its range is [0, ∞). x y
x y 3 2 1 4
(3, 0)
f ( x ) = | x – 3 |
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9. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 2
Applying a Vertical Shift 1 x
Graph f ( x ) = – 4.
This graph of this function is found by shifting
f ( x ) = four units down. x y 1 x
x y –1 1 3 –2 2 –3
–3.5
x y 7 1 3 6 2 5 –
4.5
f ( x ) = – 4 1 x
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10. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 2
Applying a Vertical Shift 1 x
Graph f ( x ) = – 4.
This graph of this function is found by shifting
f ( x ) = four units down. x y 1 x
The domain is (– ∞, 0) U (0, ∞) and the
range is (– ∞, –4) U (–4, ∞).
Notice the vertical asymptote at x = 0
and the horizontal asymptote at y = –4.
f ( x ) = – 4 1 x
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11. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 3
Applying Both Horizontal and Vertical Shifts
Graph f ( x ) = – 3.
x + 2
This graph is found by shifting f ( x ) = two units to the left and
three units down. x
The domain of this function is [–2, ∞) and its range is [–3, ∞). x y
x y –3 –2 –1 2 7
f ( x ) = – 3
x + 2
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12. 12.1 Additional Graphs of Functions; Composition
Greatest Integer Function
The greatest integer function, usually written f (x) = x , is defined as
follows:
x denotes the largest integer that is less than or equal to x.
For example,
9 = 9, – = –4, = 5.
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13. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 4
Graphing the Greatest Integer Function
Graph f ( x ) = x .
For x , x y
if –1 ≤ x < 0, then x = –1;
f ( x ) = x
if ≤ x < 1, then x = 0;
if ≤ x < 2, then x = 1,
and so on.
The appearance of the graph is the
reason that this function is called a
step function.
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14. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 5
Applying a Greatest Integer Function
An overnight delivery service charges $20 for a package weighing up to 4lb.
For each additional pound or fraction of a pound there is an additional charge
of $2. Let D(x) represent the cost to
send a package weighing x pounds.
Graph D(x) for x in the interval (0, 7]. y x
Dollars 10 20 30 1 2 3 4 5 6 7
Pounds
For x in the interval (0, 4], y = 20.
For x in the interval (4, 5], y = 22.
For x in the interval (5, 6], y = 24.
For x in the interval (6, 7], y = 26,
and so on.
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15. 12.1 Additional Graphs of Functions; Composition
Composition of Functions
Composition of Functions
If f and g are functions, then the composite function, or composition,
of g and f is defined by
( g ◦ f )( x ) = g ( f ( x ) )
for all x in the domain of f such that f (x) is in the domain of g.
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16. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 6
Evaluating a Composite Function
Let f (x) = 3x2 + 5 and g(x) = x – 7. Find ( f ◦ g )( 2 ).
( f ◦ g )( 2 ) = f ( g ( 2 ) ) Definition
= f ( 2 – 7 )
Use the rule for g( x ); g( 2 ) = 2 – 7.
= f ( –5 )
Subtract.
= 3( –5 )2 + 5
Use the rule for f ( x ); f ( –5 ) = 3( –5 )2 + 5.
= 80
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17. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 6
Evaluating a Composite Function
Let f (x) = 3x2 + 5 and g(x) = x – 7. Now find ( g ◦ f )( 2 ).
( g ◦ f )( 2 ) = g ( f ( 2 ) ) Definition
= g ( 3( 2 ) )
Use the rule for f ( x ); f ( 2 ) = 3( 2 )2 + 5.
= g ( 17 )
Square, multiply, and then add.
= 17 – 7
Use the rule for g( x ); g(17) = 17 – 7.
= 10
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18. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 6
Evaluating a Composite Function
Let f (x) = 3x2 + 5 and g(x) = x – 7. Notice that ( f ◦ g )( 2 ) ≠ ( g ◦ f )( 2 ).
( f ◦ g )( 2 ) = f [ g ( 2 ) ]
= f ( 2 – 7 )
= f ( –5 )
= 3( –5 )2 + 5
= 80
( g ◦ f )( 2 ) = g [ f ( 2 ) ]
= g ( 3( 2 ) )
= g ( 17 )
= 17 – 7
= 10
In general, ( f ◦ g )( 2 ) ≠ ( g ◦ f )( 2 ).
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19. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 7
Finding Composite Functions
Let f (x) = 5x + 1 and g(x) = x2 – 4. Find each of the following.
(a) ( f ◦ g )( –3 )
( f ◦ g )( –3 ) = f [ g ( –3 ) ]
= f ( (–3)2 – 4 )
g(x) = x2 – 4
= f ( 5 )
= 5( 5 ) + 1
f (x) = 5x + 1
= 26
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20. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 7
Finding Composite Functions
Let f (x) = 5x + 1 and g(x) = x2 – 4. Find each of the following.
(b) ( f ◦ g )( n )
( f ◦ g )( n ) = f [ g ( n ) ]
= f ( n2 – 4 )
g(x) = x2 – 4
= 5(n2 – 4) + 1
f (x) = 5x + 1
= 5n2 – 19
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21. 12.1 Additional Graphs of Functions; Composition
EXAMPLE 7
Finding Composite Functions
Let f (x) = 5x + 1 and g(x) = x2 – 4. Find each of the following.
(c) ( g ◦ f )( n )
( g ◦ f )( n ) = g [ f ( n ) ]
= g ( 5n )
f (x) = 5x + 1
= (5n + 1)2 – 4
g(x) = x2 – 4
= 25n n – 4
= 25n n – 3
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