Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Functions(Continued)
March 21, 2021
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Outline
1 Algebra of Functions
2 One-to-one Function
3 The inverse of a Function
4 Graphing Functions and their Inverses
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Sum, Difference, Product and Quotient of functions
Let f and g be functions. Then
1. the sum, f + g , is the function defined by
(f + g )(x) = f (x) + g (x).
2. the difference, f − g , is the function defined by
(f − g )(x) = f (x) − g (x).
3. the product, f · g , is the function defined by
(f · g )(x) = f (x) · g (x).
f
4. the quotient, , is the function defined by
g
f f (x)
(x) = , g (x) 6= 0.
g g (x)
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
The domains of f + g , f − g , and f · g consist of the numbers
x in the domains of both f and g i.e domain of f ∩ domain of
g.
f
The domain of is the set of numbers x in the domains of
g
both f and g with g (x) 6= 0.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 1
Let f and g be two functions defined by
1
f (x) = 2x + 1, g (x) = .
3x − 2
f
Find (i) f + g , (ii) f − g , (iii) f · g , and (iv) g.
Solution:
(i)
1
(f + g )(x) = f (x) + g (x) = (2x + 1) +
3x − 2
(2x + 1)(3x − 2) + 1
=
3x − 2
6x 2 − x − 1
=
3x − 2
Domain of f + g = {x ∈ R : x 6= 23 }
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
(ii)
1
(f − g )(x) = f (x) − g (x) = (2x + 1) −
3x − 2
(2x + 1)(3x − 2) − 1
=
3x − 2
2
6x − x − 3
=
3x − 2
Domain of f − g = {x ∈ R : x 6= 23 }
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
(iii)
1
(f · g )(x) = f (x) · g (x) = (2x + 1)( )
3x − 2
2x + 1
=
3x − 2
Domain of f · g = {x ∈ R : x 6= 32 }
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
(iv)
f f (x) 2x + 1
( )(x) = = 1
g g (x) 3x−2
= (2x + 1)(3x − 2)
= 6x 2 − x − 2
f
Domain of = {x ∈ R : x 6= 23 }
g
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
One-to-one Function
A function f with domain A is called a one-to-one function if no
two elements of A have the same image; that is
f (x1 ) 6= f (x2 ) whenever x1 6= x2 .
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Horizontal line test
A function is one-to-one if and only if no horizontal line intersects
its graph more than once.
Figure: one-to-one
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Horizontal line test
Figure: Not one-to-one
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 2
1. Linear functions, f (x) = ax + b, a 6= 0, is a one-to-one
function.
2. The function g (x) = x 2 , x ∈ R is not one-to-one because
g (2) = 22 = 4 and g (−2) = (−2)2 = 4
and so 2 and -2 have the same image.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Remark
Although the function g = x 2 in Example 2 is not one-to-one, it is
possible to restrict its domain so that the resulting function is
one-to-one. In fact, if we define
h(x) = x 2 , x ≥ 0,
then h is one-to-one.
Figure: h(x) = x 2 , x ≥ 0 is one-to-one
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
The inverse of a function
Let f be a one-to-one function with domain A and range B. Then
a function f −1 with domain B and range A is called the inverse
function of f if one has
1. (f ◦ f −1 )(x) = f (f −1 (x)) = x for all x ∈ B, and
2. (f −1 ◦ f )(x) = f −1 (f (x)) = x for all x ∈ A.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 3
x −4
Show that the functions f (x) = 3x + 4 and g (x) = are
3
inverses of each other.
Solution:
x − 4 x − 4
f (g (x)) = f =3 +4=x −4+4=x
3 3
3x + 4 − 4 3x
g (f (x)) = g (3x + 4) = = = x.
3 3
Hence, f and g are inverses of each other.
Note:
1
f −1 (x) 6= = [f (x)]−1
f (x)
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Finding the inverse
The procedure for finding the inverse of a one-to-one function is
outlined below,
1. Write y = f (x).
2. Solve this equation for x in terms of y .
3. Interchange x and y . The resulting equation is y = f −1 (x).
Note: We want to emphasize that
Domain of f −1 = range of f
range of f −1 = domain of f
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 4
Determine the inverse function of the following functions.
1. f (x) = 7 − 5x, x ∈ R.
2x + 5
2. f (x) = , x 6= 7.
x −7
√
3. f (x) = 2 + 3 + x, x ≥ −3
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
1. f (x) = 7 − 5x, x ∈ R.
First we write y = f (x).
y = 7 − 5x
Then we solve for x:
y = 7 − 5x
5x = 7 − y
7−y
x=
5
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
Finally, we interchange x and y :
7−x
y=
5
Hence
7−x
f −1 (x) = , x ∈R
5
2x + 5
2. Let y = , x 6= 7. Then
x −7
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
y (x − 7) = 2x + 5
xy − 7y = 2x + 5
xy − 2x = 7y + 5
x(y − 2) = 7y + 5
7y + 5
x=
y −2
Therefore,
7x + 5
f −1 (x) = , x 6= 2.
x −2
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
√
Let y = 2 + 3 + x, x ≥ −3. Then
√
y −2= 3+x
(y − 2)2 = 3 + x (squaring both sides)
x = (y − 2)2 − 3
f −1 (x) = (x − 2)2 − 3
√
Note that for x ≥ −3, x + 3 ≥ 0 and so x + 3 ≥ 0. Therefore
√
y =2+ 3+x ≥2
Hence, the range of f is y ≥ 2. It means that the domain of f −1 is
x ≥ 2. That is,
f −1 (x) = (x − 2)2 − 3, x ≥ 2
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 5
Prove that the function
1−x
f (x) =
1+x
is inverse to itself.
Solution To show that f (x) is self inverse, we show that
f (f (x)) = x. To see this,
1−x
1−x 1− 1+x
f (f (x)) = f =
1+x 1−x
1+
1+x
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution
1 + x − (1 − x)
= 1+x
1+x +1−x
1+x
2x
= 1+x
2
1+x
2x 1+x
= · =x
1+x 2
Hence, f is inverse to itself.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Graphing Functions and their Inverses
Let y = f (x) be a one-to-one function. If f (a) = b, then
f −1 (b) = a. Therefore, a point (a, b) is on the graph of y = f (x)
if and only if (b, a) is on the graph of y = f −1 (x).
Looking at the figure above, we see that (b, a) is obtained from
(a, b) by reflection in the line y = x.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Graphing Functions and their Inverses
Therefore, the graph of f −1 is obtained by reflecting the graph of
f in the line y = x.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 6
Let f (x) = 2x + 5, x ∈ R, be a function.
i. Find the inverse function, f −1 , of f .
ii. Sketch the graphs of y = f (x) and y = f −1 (x).
Solution:
y −5
i. Let y = 2x + 5. Then x = . Therefore,
2
x −5
f −1 (x) = , x ∈R
2
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Solution (ii)
From the graph above, we observe that that the graph of y = f −1
is a reflection of y = f (x) in the line y = x.
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Example 7
√
Let f (x) = x − 2.
i. Sketch the graph of y = f (x) and obtain the graph of
y = f −1 (x) from the graph of y = f (x).
ii. Find f −1 (x).
Solution:
√
i. Given that f (x) = x − 2. First observe that the domain of f
is x ≥ 2. √
When x = 2, f (2) = √2 − 2 = √0
When x = 6, f (6) = 6√− 2 = 4 √ =2
When x = 11, f (11) = √11 − 2 = √9 = 3
When x = 18, f (18) = 18 − 2 = 16 = 4
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Functions(Continued)
Outline
Algebra of Functions
One-to-one Function
The inverse of a Function
Graphing Functions and their Inverses
Exercise
1. Complete example 6(ii)
2. Let g (x) = x 2 + 1, x ≥ 0.
(i) Sketch the graph of f . (ii) Obtain the graph of f −1 from f
and (iii) find f −1 .
x 1
3. Let f (x) = and g (x) = . Find f ◦ g and its domain.
x +1 x
4. Suppose that g (x) = 2x + 1 and h(x) = 4x 2 + 4x + 7. Find a
function f such that f ◦ g = h.
Functions(Continued)
