Unformatted text preview: MAC 1105
Module 6
Composite Functions and
Inverse Functions
Rev.S08 Learning Objectives
Upon completing this module, you should be able to:
1.
2.
3.
4.
5.
6.
7. Perform arithmetic operations on functions.
Perform composition of functions.
Calculate inverse operations.
Identity onetoone functions.
Use horizontal line test to determine if a graph represents a
onetoone function.
Find inverse functions symbolically.
Use other representations to find inverse functions. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 2 Composite Functions and
Inverse Functions
There are two major topics in this module:  Combining Functions; Composite Functions
Inverse Functions Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 3 Five Ways of Combining Functions (g=)g
))f()g
f+)fx()
g=()
( (+
xxx
x
())f()g
fxx()
(×
xx
(öf()w)¹0
fgx
()
=
f÷ x
æ ghere
()
=
çxg (
gx(())
è()fg
ø=x
(g
)
fo If f(x) and g(x) both exist, the sum, difference,
If
and
both
product, quotient and composition of two functions f
and g are defined by Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 4 Example of Addition of Functions
Let f(x) = x2 + 2x and g(x) = 3x  1
Find the symbolic representation for the function f + g
Find
and use this to evaluate (f + g)(2).
and
(f + g)(x) = (x2 + 2x) + (3x − 1)
(f + g)(x) = x2 + 5x − 1
(f + g)(2) = 22 + 5(2) − 1 = 13
5(
or (f + g)(2) = f(2) + g(2)
)(2) (2)
(2)
= 22 + 2(2) + 3(2) − 1
= 13
13 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 5 Example of Subtraction of Functions
Let f(x) = x2 + 2x and g(x) = 3x − 1
Find the symbolic representation for the
function f − g and use this to evaluate
(f − g)(2).
(f − g)(x) = (x2 + 2x) − (3x − 1)
3x
(f − g)(x) = x2 − x + 1
So (f − g)(2) = 22 − 2 + 1 = 3
So
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 6 Example of Multiplication of Functions
Let f(x) = x2 + 2x and g(x) = 3x − 1
Find the symbolic representation for the
function fg and use this to evaluate (fg)(2)
(fg)(x) = (x2 + 2x)(3x − 1)
(fg)(x) = 3x3 + 6x2 − x2 − 2x
(fg)(x) = 3x3 + 5x2 − 2x
So (fg)(2) = 3(2)3 +5(2)2 − 2(2) = 40
So
3(2) +5(2)
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 7 Example of Division of Functions
Let f(x) = x2 + 2x and g(x) = 3x − 1
Find the symbolic representation for the function
Find
and use this to evaluate So
So
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 8 Example of Composition of Functions
Let f(x) = x2 + 2x and g(x) = 3x  1
Find the symbolic representation for the function
f ° g and use this to evaluate (f ° g)(2)
)(2)
(f ° g)(x) = f(g(x)) = f(3x – 1) = (3x – 1)2 + 2(3x – 1)
))
(f ° g)(x) = (3x – 1) ( 3x – 1) + 6x – 2
(f ° g)(x) = 9x2 – 3x – 3x + 1 + 6x – 2
(f ° g)(x) = 9x2 – 1
So (f ° g)(2) = 9(2)2 – 1 = 35
35
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 How to Evaluate Combining of Functions
Numerically?
Given numerical
representations for f
and g in the table Evaluate combinations
of f and g as
specified.
specified.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 10 How to Evaluate Combining of Functions
Numerically? (Cont.)
(f + g)(5) = f(5) + g(5) = 8 + 6 = 14
14
(fg)(5) = f(5) • g(5) = 8 • 6 = 48
48
(f ° g)(5) = f(g(5)) = f(6) = 7 Try to work out the rest of
them now.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 11 How to Evaluate Combining of Functions
Numerically? (Cont.)
Check your answers: Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 12 How to Evaluate Combining of Functions
Graphically?
Use graph of f and g below to evaluate
(f + g) (1) y = f(x) (f – g) (1)
(f • g) (1)
(f/g) (1)
(f ° g) (1)
Can you identify the two functions?
Try to evaluate them now.
Hint: Look at the yvalue when x = 1.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. y = g(x)
13 How to Evaluate Combining of Functions
Graphically?
y = f(x)
Check your answer now. (f + g) (1) = f(1) + g(1) = 3 + 0 = 3
(f – g) (1) = f(1) – g(1) = 3 – 0 = 3 y = g(x) (fg) (1) = f(1) • g(1) = 3 • 0 = 0
(f/g) (1) is undefined, because division by 0 is undefined.
(f ° g) (1) = f(g(1)) = f(0) = 2
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 14 Next, Let’s Look at Inverse Functions
and Their Representations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 15 A Quick Review on Function
• y = f(x) means that given an input x, there
is just one corresponding output y.
• Graphically, this means that the graph
passes the vertical line test.
• Numerically, this means that in a table of
values for y = f(x) there are no xvalues
repeated. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 16 A Quick Example
Given y2 = x, iis y = f(x)? That is, is y a function of
s
)?
x?
No, because if x = 4, y could be 2 or –2.
No,
Note that the graph fails the vertical line test.
x y 4 –2 1 –1 0 0 1 1 4 2 Note that there is a value of x in the table for which
Note
there are two different values of y (that is, xvalues
values
are repeated.)
are Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 17 What is OnetoOne?
• Given a function y = f(x), f is onetoone
means that given an output y there was just
one input x which produced that output.
• Graphically, this means that the graph passes
the horizontal line test. Every horizontal line
intersects the graph at most once.
• Numerically, this means the there are no yvalues repeated in a table of values. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 18 Example
• Given y = f(x) = x, is f onetoone?
Given
,
onetoone
– No, because if y = 2, x could be 2 or – 2.
No,
• Note that the graph fails the horizontal line test.
Note
horizontal
x y –2 2 –1 1 0 0 1 1 2 2• Rev.S08 (2,2) (2,2) Note that there is a value of y in the table for
Note
which there are two different values of x (that is,
yvalues are repeated.)
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 19 What is the Definition of a
OnetoOne Function?
A function f is a onetoone function if, for elements c and d in
the domain of f,
c ≠ d implies f(c) ≠ f(d)
Example: Given y = f(x) = x, f is not onetoone because –2
≠ 2 yet  –2  =  2  Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 20 What is an Inverse Function?
f 1 is a symbol for the inverse of the function f, not to be
confused with the reciprocal.
If f 1(x) does NOT mean 1/ f(x), what does it mean?
y = f 1(x) means that x = f(y)
Note that y = f 1(x) is pronounced “y equals f inverse of x.” Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 21 Example of an Inverse Function
Let F be Fahrenheit temperature and let C be Centigrade
temperature.
F = f(C) = (9/5)C + 32
C = f 1(F) = ?????
• The function f multiplies an input C by 9/5 and adds 32.
• To undo multiplying by 9/5 and adding 32, one should
subtract 32 and divide by 9/5
So C = f 1(F) = (F – 32)/(9/5)
= (5/9)(F – 32) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 22 Example of an Inverse Function (Cont.)
F = f(C) = (9/5)C + 32
C = f 1(F) = (5/9)(F – 32)
• Evaluate f(0) and interpret.
f(0) = (9/5)(0) + 32 = 32
When the Centigrade temperature is 0, the Fahrenheit
temperature is 32.
• Evaluate f 1(32) and interpret.
f 1(32) = (5/9)(32  32) = 0
When the Fahrenheit temperature is 32, the Centigrade
temperature is 0.
Note that f(0) = 32 and f 1(32) = 0
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 23 Graph of Functions and Their Inverses
1
• The graph of f 1 is a reflection of the graph of
reflection
f across the line y = x 1
Note that the domain of f equals the range of f 1 and the
Note
1
range of f equals the domain of f 1 . Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 How to Find Inverse Function
Symbolically?
• 1
Check that f is a onetoone function. If not, f 1 does not
Check
onetoone
If
exist.
exist.
•
Solve the equation y = f(x) for x, resulting in the
Solve
for resulting
1
1(y)
equation x = f
1
•
Interchange x and y to obtain y = f 1(x)
Interchange
Example.
Example. –
–
–
–
–
– Rev.S08 Step 1  Is this a onetoone function? Yes. f(x) = 3x + 2
Step
Step 2  Replace f(x) with y: y = 3x + 2
Step
f(x)
Step 3  Solve for x: 3x = y – 2
x:
x = (y – 2)/3
2)/3
Step 4  Interchange x and y: y = (x – 2)/3
y:
1
1
Step 5  Replace y with f 1(x): So f 1(x) = (x – 2)/3
Step
x): http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 25 How to Evaluate Inverse Function
Numerically? x f(x)
1 –5
2 –3
3
0
4
3
5
5
Rev.S08 1
The function is onetoone, so f 1
onetoone
exists.
exists.
1
f 1(–5) = 1
1
f 1(–3) = 2
1
f 1(0) = 3
1
f 1(3) = 4
1
f 1(5) = 5
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 26 How to Evaluate Inverse Function
Graphically?
• The graph of f below
passes the horizontal
line test so f is onetoline
onetoone.
• Evaluate f 1(4).
Evaluate 1
• Since the point (2,4) is
Since
on the graph of f, the
the
point (4,2) will be on the
1
graph of f 1 and thus
1
f 1(4) = 2
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. f(2)=4 27 What is the Formal Definition of
Inverse Functions?
1
Let f be a onetoone function. Then f 1 is the
inverse function of f, iif
f
1
1
• (f 1 o f)(x) = f 1(f(x)) = x for every x in the domain
))
of f
1
1
• (f o f 1 )(x) = f(f 1(x)) = x for every x in the domain
))
1
of f 1 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 28 What have we learned?
We have learned to:
1.
2.
3.
4.
5.
6.
7. Perform arithmetic operations on functions.
Perform composition of functions.
Calculate inverse operations.
Identity onetoone functions.
Use horizontal line test to determine if a graph represents a
onetoone function.
Find inverse functions symbolically.
Use other representations to find inverse functions. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 29 Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th
Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 30 ...
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Full Document
 Winter '08
 Staff
 Algebra, Inverse Functions, Composite Functions, Inverse function, Injective function, Function composition, Click Link, Subtraction of Functions

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