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© The Visual Classroom
1.5 Inverse Functions
Example 1: An equation for determining the cost of a taxi ride is: C = 0.75n + 2.50, where n is the number of kilometers and C is the cost.
n C10
20
30
40
$10.00
$17.50
$25.00
$32.50
Complete the table of values.
© The Visual Classroom
The inverse function allows us to ask the question: if I have $50, how far can I travel?
original function: C = 0.75n + 2.50
× 0.75 + 2.50n C
input number of kilometers
output Cost
inverse function
÷ 0.75 – 2.50n C
output number of kilometers
input Cost2.50
0.75
Cn
© The Visual Classroom
A Bf
6 23
24
715
Domain: {2, 4, 6}
Range: {7, 15, 23}
We know that a function maps elements of a domain onto elements of a range.
A B
6 23
24
715
f –1
The inverse function maps the elements of the range back onto the elements of the domain.
Domain: {7, 15, 23}
Range: {2, 4, 6}
© The Visual Classroom
Assume we have a function which consists of a set of ordered pairs.
f(x) = {(1, 5), (2, 9), (3, 13), (4, 17)}
Domain: {1, 2, 3, 4} Range: {5, 9, 13, 17}
f –1(x) = {(5, 1), (9, 2), (13, 3), (17, 4)}
Range: {1, 2, 3, 4}Domain: {5, 9, 13, 17}
f –1(x) means the inverse function of f(x).
© The Visual Classroom
Suppose we have the following relation f(x) consisting of the following points. Determine the graph of f –1(x).
We have symmetry about the line y = x.
f(– 4) =
f(0) =
?
?
2
5
f –1(1) = ?
f –1(0) = ?
–2
–6
© The Visual Classroom
Example 2: Given the graph of y = f(x) below, sketch the graph of y = f –1(x).
Step 1: Sketch the graph of y = x.
Step 2: Map the points using the line y = x as the axis of symmetry.
Step 3: Join the points
D: x and R: y 0
f(2) = ?8
f –1(2) = ?0
© The Visual Classroom
f –1(x)
D: x 0 and R: y
f(x)
D: x and R: y 0
State the domain and range of f –1(x)
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f(x) = 3x + 4Example 3: Given the equation:
Determine the equation of f – 1(x).
y = 3x + 4f f –1
1. Replace y by x and x by y.
2. Isolate y.
x – 4 = 3y
4
3
xy
x = 3y + 4
© The Visual Classroom
Compare f and f –1 and the order in which operations are carried out.
1. Multiply by 3
2. Add 4
1. Subtract 4
2. Divide by 3
y = 3x + 4
f f –1
4
3
xy
You will notice the order and the operations are inverted.
© The Visual Classroom
When you go to bed at night
1- you untie your laces
2- you take off your shoes 2- you put on your shoes
When you get up in the morning
1- you put on your socks
Notice the inverse operation
3- you take off your socks 3- you tie up your laces
Reverse order … reverse operation
© The Visual Classroom
1. Multiply by 5
2. Subtract 2
1. Add 2
2. Divide by 5
f(x) = 5x – 2 f –1
1 2( )
5
xf x
Example 4: Determine the inverse of f(x) = 5x – 2
x + 2
2
5
x
Ex: f(4) = 5(4) – 2= 20 – 2= 18
f –1(4) =4 2
5
6
5
© The Visual Classroom
Example 5: Given:3
( ) ( 4)2
f x x
a) Determine f –1(x)
3( 4)
2x y
2 3( 4)x y
2 3 12x y
2 12
3
xy
1 2 12
( )3
xf x
replace x by y and y by x.
× 2
isolate y.
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1 2 12( )
3
xf x
3
( ) ( 4)2
f x x
1 2( 1) 12( 1)
3
bf b
2 2 12
3
b
2 14
3
b
b) Determine f (–6)
3( 6) ( 6 4)
2f
3( 10)
2
15
c) Determine f –1(b + 1)
© The Visual Classroom
Example 6: the relation f is defined by 2x – 3y = 6.
Graph f
x f(x)
Using the intercept method.
0 – 203
Graph f –1
f -1(x)x
0– 20 3
f
f –1
© The Visual Classroom
The relation f is defined by 2x – 3y = 6.
f
f –1
Determine:
f(–3) = – 4
f –1(–3) = – 1.5
f –1(x):
2y – 3x = 6
2y = 3x + 6
33
2y x