FINDING THE INVERSE OF AN ELEMENT IF THE FUNCTION IS GIVEN

Question 1 :

If f -> Q -> Q is given by f(x)  =  x², then find

(i)  f^-1 (9)  (ii)  f^-1(-5)  (iii)  f^-1(0)

Solution :

From the given function, we have

 f(x)  =  x²

x  =  f^-1(x²)  ----(1)

(i)  f^-1 (9)

By comparing f^-1 (9) with (1), we get that 

x²  =  9

x  =  ±3

(ii)  f^-1(-5)  

By comparing f^-1 (5) with (1), we get that 

x²  = -5

x  =  √-5 which is not possible

f^-1(-5)  is undefined.

(iii)  f^-1(0)

By comparing f^-1 (0) with (1), we get that 

x²  =  0

x  =  0

Hence the value of f^-1(0)   =  0.

Question 2 :

In the function f : R-> R be defined by f(x)  =  x² + 5x + 9, find f^-1 (8) and f^-1(9)

Solution :

Given that :  f(x)  =  x² + 5x + 9

x  =  f^-1(x² + 5x + 9)  ---(1)

f^-1 (8)  ---(2)

By comparing (1) and (2), we have 8 instead of x² + 5x + 9

x² + 5x + 9  =  8

x² + 5x + 9 - 8  =  0

x² + 5x + 1  =  0

x  =  (-b±√b² - 4ac)/2a

x  =  (-5±√(5²-4))/2(1)

x  =  (-5±√21)/2

f^-1(9)

x² + 5x + 9  =  9

x² + 5x + 9 - 9  =  0

x² + 5x  =  0

x(x + 5)  =  0

x  =  0 and x  =  -5

Question 3 :

Let R-> R be defined on f(x)  =  x² + 1. Find 

(i)  f^-1 (-5)  (ii)  f^-1(26)  (iii)  f^-1[10, 37]

Solution :

f(x)  =  x² + 1

(i)  f^-1 (-5)  

x² + 1  =  -5

x²  =  -6 Which is not real.

(ii)  f^-1(26)

x² + 1  =  26

x²  =  26 - 1

x²  =  25

x  =  ± 5

(iii)  f^-1[10, 37]

Hence the values of x are [-6, -3, 3, 6].

Question 4 :

Find the inverse of the relation.

(1, 0), (3, −8), (4, −3), (7, −5), (9, −1)

Solution :

Inverse relation is 

(0, 1) (-8, 3) (-3, 4) (-5, 7) (-1, 9)

Question 5 :

Find the inverse of the relation.

(2, 1), (4, −3), (6, 7), (8, 1), (10, −4)

Solution : 

The inverse relation is,

(1, 2) (-3, 4) (7, 6) (1, 8) (-4, 10)

Question 6 :

Solution : 

Writing the ordered pairs from the table, we get

(-5, 8) (-5, 6) (0, 0) (5, 6) (10, 8)

The inverse relation is,

(8, -5) (6, -5) (0, 0) (6, 5) (8, 10)

Use the tables below to find the given values.

Question 7 :

Solution :

a. 𝑓(1) 

When the input is 1, the output is -2.

b. 𝑓(6)

When the input is 6, the output is 1.

c. 𝑓^-1 (1)

When the output is 1, the input is 6.

d. 𝑓^-1 (4)

When the output is 4, the input is 5.

e. 𝑓 (2)

When the input is 2, the output is 3.

f.  𝑓^-1 (6)

When the input is 6, the output is 3.

Question 8 :

Solution :

a. 𝑓(2) :

When the input is 2, the output is 10.

b. 𝑓(10) :

When the input is 10, the output is 8.

c. 𝑓^-1 (7) :

When the output is 7, the input is -1.

d. 𝑓^-1 (-3) :

When the output is -3, the input is 7.

e. 𝑓(7) :

When the input is 7, the output is -3.

f. 𝑓^-1 (2) :

When the output is 2, the input is -3.

Question 10 :

Solution :

a. 𝑓(-10) :

When the input is -10, the output is -6.

b. 𝑓(3) :

When the input is 3, the output is 11.

c. 𝑓^-1 (7) :

When the input is 7, the output is 3.

d. 𝑓^-1 (-6) :

When the output is -6, the input is -10.

e. 𝑓(7) :

When the input is 7, the output is 3.

f. 𝑓^-1 (-2) :

When the output is -2, the input is 11.

Question 11 :

The graph of the piecewise-linear function 𝑓 is shown in the figure. Let 𝑔 be the inverse function of 𝑓. What is the minimum value of 𝑔?

a) -4   b)  -3    c) 2    d)  3

Solution :

Functions f and g are inverse to each other. The minimum value of f from the above graph is (3, -4)

Set of ordered pairs from graph of f are,

(-3, 2) (-1, 1) (1, -2) (3, -4)

Set of ordered pairs from graph of g are,

(2, -3) (1, -1) (-2, 1) (-4, 3)

At (2, -3) we have the minimum point. So, the answer is -3.

Question 12 :

Mr. Brust is filling up his backdoor kiddie pool with the water hose. The amount of water, in gallons, in the pool 𝑡 minutes after he turns on the water can be modeled by 𝑃, an increasing function of time 𝑡. Which of the following gives a verbal representation of the function 𝑃^-1 , the inverse of 𝑃? 

a) 𝑃^-1 is an increasing function of the amount of time after the water is turned on.

b) 𝑃^-1 is a decreasing function of the amount of time after the water is turned on.

c) 𝑃^-1 is an increasing function of the amount of water in the pool.

d) 𝑃^-1 is a decreasing function of the amount of water in the pool.

Solution :

Accordingly the function P :

• Dependent variable = time
• Independent variable = Amount of water in gallons.

Accordingly inverse function of  𝑃^-1 :

• Independent variable = time
• Dependent variable = Amount of water in gallons.

So, option a is correct.

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