FIND INVERSE FUNCTION FROM THE GIVEN FUNCTION

For each of the following functions :

Example 1 :

a) f(x)  =  2x+5     b) f(x)  =  (3-2x)/4     c) f(x)  =  x+3

(i) Find f^-1(x)

(ii) sketch y  =  f(x), y  =  f^-1(x) and y  =  x on the same axes.

Solution :

a) f(x)  =  2x+5

(i)

Step 1 :

Let, y  =  f(x)

y  =  2x+5

Step 2 :

By interchanging x and y, we get

x  =  2y+5

2y  =  x-5

y  =  (x-5)/2

Step 3 :

So, the required inverse function is

f^-1(x)  =  (x-5)/2

(ii)  By applying some random values of x in the function

y  =  2x+5

By joining the points (-1, 3), (0, 5) and (1, 7), we will get the graph of original function. So, the required coordinates of inverse function are (3,-1) (5, 0) and (7, 1).

We draw the graph of inverse function be reflecting the graph of original function about y = x.

b)  f(x)  =  (3-2x)/4

Solution :

(i)  Let, y  =  f(x)

y  =  (3-2x)/4

By interchanging x and y, we get

x  =  (3-2y)/4

4x  =  3-2y

2y  =  3-4x

y  =  (3-4x)/2

y  =  -2x+(3/2)

We express f^-1 as a function of x in terms of y

f^-1(x)  =  -2x+(3/2)

(ii)  For graphing, by applying some random values of x in the function

 y  =  -2x+(3/2)

By joining the points (-1, 5/4), (0, 3/4) and (1, 1/4), we will get the graph of original function. So, the required coordinates of inverse function are (5/4, -1), (3/4, 0) and (1/4, 1)

By plotting the points, we get graph of the given function and inverse function.

c) f(x)  =  x+3

Solution :

(i)  Let, y  =  f(x)

y  =  x+3

By interchanging x and y, we get

x  =  y+3

y  =  x-3

We express f^-1 as a function of x in terms of y

f^-1(x)  =  x-3

(ii) 

Coordinates of given function : (-1, -4) (0, -3) and (1, -2) Coordinates of inverse function : (-4, -1) (-3, 0) and (-2, 1)

Example 2 :

If f(x)  =  2x + 7, find :

a) f^-1(x)        b) f(f^-1(x))        c) f^-1(f(x))

Solution :

a.

Given, f(x)  =  2x + 7

Let y  =  2x+7

By interchanging x and y, we get

x  =  2y+7

2y  =  x-7

y  =  (x-7)/2

We express f^-1 as a function of x in terms of y

f^-1(x)  =  (x-7)/2

b)

We have, f^-1(x)  =  (x-7)/2

Then,

f(f^-1(x))  =  f((x-7)/2)

=  2x+7

=  2[(x-7)/2] + 7

=  x–7+7

f(f^-1(x))  =  x   ----(1)

c)

we have, f^-1(x)  =  (x-7)/2

f^-1(f(x))  =  (f(x) - 7)/2

=  (2x+7-7)/2

=  2x/2

f^-1(f(x))  =  x ----(2)

Example 3 :

If f(x)  =  (2x + 1)/(x + 3), find :

a) f^-1(x)     b) f(f^-1(x))     c) f^-1(f(x))

Solution :

a) 

Given, f(x)  =  (2x+1)/(x+3)

Let y  =  (2x+1)/(x+3)

By interchanging x and y, we get

x  =  (2y+1)/(y+3)

x(y+3)  =  2y+1

xy+3x  =  2y+1

-2y+xy  =  -3x+1

y(-2+x)  =  -3x+1

y  =  (-3x+1)/(x-2)

y  =  (1-3x)/(x-2)

We express f^-1 as a function of x in terms of y

f^-1(x)  =  (1-3x)/(x-2)

b)  f(f^-1(x))  =  f((1-3x)/(x-2))

By using x in terms of f(x), we get

f(x)  =  (2x+1)/(x+3)

=  [2[(1-3x)/(x-2)]+1]/[(1-3x)/(x-2)+3]

=  [(2-6x+x-2)/(x-2)]/[(1-3x+3x-6)/(x-2)]

=  (-5x)/(-5)

=  x

f(f^-1(x))  =  x  ------(1)

c)  we have, f^-1(x)  =  (1-3x)/(x-2)

By using f(x) in terms of f^-1(x), we get

f^-1(f(x))  =  (1-3f(x))/(f(x)-2)

simplifying the terms,

=  [1-3(2x+1)/(x+3)]/[(2x+1)/(x+3)-2)

=  [(x+3) – 3(2x+1)/(x+3)]/[(2x+1) – 2(x+3)/(x+3)]

=  [x+3–6x–3]/[2x+1–2x–6]

=  (-5x)/(-5)

=  x

f^-1(f(x))  =  x  ------(2)

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