inverse functions
Module 3 : Investigation 5
MAT 170 | Precalculus
September 16, 2016
question 1
The formula that determines the perimeter p of a square (in inches)
in terms of the length (in inches) of the side of the square s is p = 4s.
(a) Define a function f that determines the perimeter of a square f(s)
in terms of the square’s side length.
(c) Define a function h that determines a square’s side length h(p) in
terms of the square’s perimeter.
(d) Use function composition to evaluate :
(i) h(f(2.5))
(ii) f(h(10))
(iii) h(f(7))
(iv) f(h(28))
(e) How are the functions f and h related ? How are the input and
output quantities of f and h related ?
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question 1 - solutions
The formula that determines the perimeter p of a square (in inches)
in terms of the length (in inches) of the side of the square s is p = 4s.
(a) Define a function f that determines the perimeter of a square f(s)
in terms of the square’s side length.
f(s) = 4s
(c) Define a function h that determines a square’s side length h(p) in
terms of the square’s perimeter.
h(p) =
p
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(d) Use function composition to evaluate :
(i) h(f(2.5)) = h(10) = 2.5
(ii) f(h(10)) = f(2.5) = 10
(iii) h(f(7)) = h(28) = 7
(iv) f(h(28)) = f(7) = 28
3
question 1 - solutions
(e) How are the functions f and h related ? How are the input and
output quantities of f and h related ?
The function h “undoes” the function f, and vice versa.
The input to f (square’s side length) is the output of h.
The input to h (square’s perimeter) is the output of f.
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definition of an inverse function
Suppose we have a function f taking an input of x to the output of
f(x).
f
x
f(x)
g
Domain
Range
If the function g takes each of the outputs f(x) to the corresponding
input x, we call g the inverse of f.
Think - “g undoes f ”
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definition of an inverse function
Definition : Inverse Function
A function g is said to be the inverse of a function f if :
(1) (f ◦ g)(x) = f(g(x)) = x for all x in the domain of g
(2) (g ◦ f)(x) = g(f(x)) = x for all x in the domain of f
We usually denote the inverse of a function f by f −1 .
Warning : f −1 ̸=
1
f
The inverse of a function does not always exist.
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when does the inverse of function exist ?
question 8*
Below is the graph of the function f(x) = x2 .
4.8
4
3.2
2.4
1.6
0.8
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
4.8
-0.8
Could the function f have an inverse ? If so, determine f −1 . If not,
explain why not.
No. f−1 is not a function since, for example, f−1 (4) = 2 and −2.
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one-to-one function
In order for f −1 to exist, the function f must be a one-to-one
function. This means that for every output f(x), there is a unique
input x.
A simple way to check if a function is one-to-one is the horizontal
line test :
The Horizontal Line Test : A function f is one-to-one if every horizontal line intersects the graph of f in at most 1 point.
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one-to-one function
4.8
4.8
4
4
3.2
3.2
2.4
2.4
1.6
1.6
0.8
0.8
-3.2
-2.4
-1.6
-0.8
0
-3.2
0.8
-2.4
1.6
-1.6
2.4
-0.8
3.2
0
4
-0.8
0.8
1.6
2.4
3.2
4
4.8
4.8
-0.8
Not one-to-one
One-to-one
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how do we find the inverse ?
question 5
The standard formula for determine temperature in degrees
Fahrenheit F when given the temperature in degrees Celsius C is
F = 95 C + 32. Consider the function g defined by
g(C) =
9
C + 32.
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(a) What does g(100) represent ?
(b) Solve the equation g(C) = 212.
(c) Define a function h that converts degrees Fahrenheit to degrees
Celsius. (Hint : Generalize the what you did in part (b))
(e) Without performing any calculations, determine the values of
g(h(212)) and h(g(100)).
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question 5 - solutions
The standard formula for determine temperature in degrees
Fahrenheit F when given the temperature in degrees Celsius C is
F = 95 C + 32. Consider the function g defined by
g(C) =
9
C + 32.
5
(a) What does g(100) represent ?
The temperature in degrees Fahrenheit corresponding to 100 degrees Celsius.
(b) Solve the equation g(C) = 212.
g(C) = 212 =⇒
9
9
C + 32 = 212 =⇒
C = 180 =⇒ C = 100
5
5
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question 5 - solutions
The standard formula for determine temperature in degrees
Fahrenheit F when given the temperature in degrees Celsius C is
F = 95 C + 32. Consider the function g defined by
g(C) =
9
C + 32.
5
(c) Define a function h that converts degrees Fahrenheit to degrees
Celsius. (Hint : Generalize the what you did in part (b))
g(C) = F =⇒
So, h(F) =
9
9
5
C+32 = F =⇒ C = F−32 =⇒ C = (F − 32)
5
5
9
5
(F − 32).
9
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question 5 - solutions
The standard formula for determine temperature in degrees
Fahrenheit F when given the temperature in degrees Celsius C is
F = 95 C + 32. Consider the function g defined by
g(C) =
9
C + 32.
5
(e) Without performing any calculations, determine the values of
g(h(212)) and h(g(100)).
g(h(212)) = 212
h(g(100)) = 100
15
question 7
Use the table to determine the following :
(a) g−1 (3) = −1
x
f(x)
g(x)
(b) g−1 (0) = 3
-2
-1
0
1
2
3
0
3
4
-1
6
-2
5
3
2
1
-1
0
(c) f(g(0) = f(2) = 6
(d) f −1 (−1) = 1
(e) g −1 (f −1 (6)) = g−1 (2) = 0
(f) f −1 (g(2)) = f−1 (−1) = 1
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