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ACTIVITY 29:
One-to-One Functions (Section 3.7, pp. 280-285)and Their Inverses
Definition of a One-One Function:
A function with domain A is called a one-to-one function if no two elements of A have the same image, that is,
f(x1) ≠ f(x2) whenever x1 ≠ x2.An equivalent way of writing the above condition is:
If f(x1) = f(x2), then x1 = x2.
Horizontal Line Test:
A function is one-to-one if and only if no horizontal line intersects its graph more than once.
Example 1:
Show that the function f(x) = 5 − 2x is one-to-one.
This line clearly passes the horizontal line test. Consequently, it is a one to one functions.
Example 2:
Graph the function f(x) = (x−2)2 −3. The function is not one-to-one: Why? Can you restrict its domain so that the resulting function is one-to-one?
The function is not one to one because it does not pass the horizontal line test.
Restricting the domain to all
x> 2 makes the function one to one.
Definition of the Inverse of a Function:Let f be a one-to-one function with domain A and range B. Then its inverse function f−1 has domain B and range A and is defined by
f−1(y) = x if and only if f(x) = y,for any y B.∈
Example 3:
Suppose f(x) is a one-to-one function.If f(2) = 7, f(3) = −1, f(5) = 18, f−1 (2) = 6 find:
f−1(7) =
f(6) =
f−1(−1) =
f(f−1(18)) =
2
2
3
f(5) = 18
If g(x) = 9 − 3x, then g−1(3) =xg )3(Let 1 )( then xg 3
339ly Consequent x9 9
63 x
36
x 2
Property of Inverse Functions:
Let f(x) be a one-to-one function with domain A and range B.The inverse function f−1(x) satisfies the following “cancellation” properties:
1.f−1(f(x)) = x for every x A ∈2.f(f−1(x)) = x for every x B∈
Conversely, any function f−1(x) satisfying the above conditions is the inverse of f(x).
Example 4:
Show that the functions f(x) = x5 and g(x) = x1/5 are inverses of each other.
))(( xgf
5
1xf
55
1
x x
))(( xfg )( 5xg 515x x
Example 5:
Show that the functions
are inverses of each other.
xxxf
2531)(
3215)(
xxxg
)(xgf
3215
xxf
321525321531
xxxx
321525321531
xxxx
32152
32325
32153
32321
xx
xx
xx
xx
32
15232532
15332
xxx
xxx
32
15232532
15332
xxx
xxx
322101510
3231532
xxx
xxx
3217
3217
x
xx
1732
3217
xxx x
How to find the Inverse of a One-to-One Function:1. Write y = f(x).2. Interchange x and y. 3. Solve this equation for x in terms of y (if possible). The
resulting equation is y = f−1(x).
Example 6:
Find the inverse of f(x) = 4x−7.
74 xy74 yxyx 47
yx
4
7
47)(1
xxf
Example 7:
Find the inverse of
21
x
y
21)(
x
xf
21
y
x
)2(2
12
yy
yx
12 yx12 xxy
xxy 21
xxy 21
xxxf 21)(1
Example 8:
Find the inverse of
22)(
xxxf
22
xxy
22
yyx
22
22
yyyyx
yyx 22
yxxy 22xyxy 22
xxy 221
122
x
xy 122)(1
xxxf
Graph of the Inverse Function:
The principle of interchanging x and y to find the inverse function also gives us a method for obtaining the graph of f−1 from the graph of f. The graph of f−1 is obtained by reflecting the graph of f in the line y = x.The picture on the right hand side shows the graphs of:
4)( xxf 0 ,4)( and 21 xxxf
Example 9:
Find the inverse of the function
xxf 11)(
Find the domain and range of f and f−1. Graph f and f−1 on the same Cartesian plane.
xy 11
yx 11
yx 11
22 11 yx
yx 11 2
yx 11 2
11)( 21 xxf
xxf 11)( 11)( 21 xxf
Domain for f(x)1x
Range for f(x)1y
Range of f-1(x)
1y
Domain of f-1(x)
1x