Find The Inverse Function from Tables Questions With Solutions

Find the values of the inverse of a function given by a table? Questions are presented along with detailed Solutions and explanations.

Relationship Between a Function and its Inverse

If f is a function whose inverse is f ^{-1}, then the relationship between f and f ^{-1} is written as:

f(a) = b ? a = f^{ -1}(b)

Examples

Use the table below to find the following if possible:
a) f^{ -1}(- 4) , b) f^{ -1}(6) , c) f^{ -1}(9) , d) f^{ -1}(10) , e) f^{ -1}(-10)

Solution

a) According to the the definition of the inverse function:
a = f^{ -1}(- 4) ? - 4 = f(a) ,
Which means that a is the value of x such f(x) = - 4.
Using the table below for x = 6, f(x) = - 4. Hence a = 6 and therefore f^{ -1}(- 4) = 6
b) a = f^{ -1}(6) ? f(a) = 6
There is no value of x for which f(x) = 6 and therefore f^{ -1}(6) is undefined.
c) a = f^{ -1}(9) ? f(a) = 9
The value of x for which f(x) = 9 is equal to - 4 and therefore f^{ -1}(9) = - 4
d) a = f^{ -1}(10) ? f(a) = 10
There is no value of x for which f(x) = 10 and therefore f^{ -1}(10) is undefined.
e) a = f^{ -1}(-10) ? f(a) = - 10
The value of x for which f(x) = -10 is equal to 8 and therefore f^{ -1}(-10) = 8

More Questions with Solutions

Use the table below to find the following if possible:
1) g^{ -1}(0) , b) g^{ -1}(-10) , c) g^{ -1}(- 5) , d) g^{ -1}(-7) , e) g^{ -1}(3)

.

Solution
a) According to the the definition of the inverse function:

a = g^{ -1}(0) ? g(a) = 0

Which means that a is the value of x such g(x) = 0.
Using the table above for x = 11, g(x) = 0. Hence a = 11 and therefore g^{ -1}(0) = 11
b) a = g^{ - 1}(- 5) if and only if g(a) = - 5
The value of x for which g(x) = - 5 is equal to 0 and therefore g^{ -1}( - 5) = 0
c) a = g^{ -1}(-10) if and only if g(a) = - 10
There is no value of x for which g(x) = -10 and therefore g^{ -1}(-10) is undefined.
d) a = g^{ -1}(- 7) ? g(a) = - 7
There is no value of x for which g(x) = - 7 and therefore g^{ -1}(- 7) is undefined.
e) a = g^{ -1}(3) if and only if g(a) = 3
The value of x for which g(x) = 3 is equal to - 2 and therefore g^{ -1}(3) = - 2