Examples on how to find inverse functions analytically are presented. Detailed solutions and matched exercises with answers at the end of this page are also included.
Example 1: Find the inverse function of f given by
f(x) = 2x + 3
Solution to example 1:
write the function as an equation.
y = 2x + 3
solve for x.
x = (y - 3)/2
now write f^{-1}(y) as follows .
f^{ -1}(y) = (y - 3)/2
or
or f^{ -1}(x) = (x - 3)/2
Check
f(f^{ -1}(x))=2(f^{ -1}(x)) + 3
=2((x-3)/2)+3
=(x-3)+3
=x
f^{ -1}(f(x))=f^{ -1}(2x+3)
=((2x+3)-3)/2
=2x/2
=x
conclusion: The inverse of function f given above is
f^{ -1}(x) = (x - 3)/2
Matched Exercise 1: Find the inverse function of f given by
f(x) = -x - 4
Example 2: Find the inverse function of f given by
f(x) = (x - 3)^{2}, if x >= 3
Solution to example 2:
write the function as an equation.
y = (x - 3)^{2}
solve for x, two solutions .
x = 3 + sqrt(y)
x = 3 - sqrt(y)
the first solution is selected
since x >= 3, write f^{-1}(y) as follows.
f^{ -1}(y) = 3 + sqrt(y)
or
f^{ -1}(x) = 3 + sqrt(x)
Check
f(f^{ -1}(x))=((3+sqrt(x))-3)^{2}
=(sqrt(x))^{2}
=x
f^{ -1}(f(x))=3+sqrt((x-3)^{2})
=3+|x-3| (since x >= 3, x-3 >= 0, |x-3| = x-3)
=3+(x-3)
=x
conclusion:
The inverse of function f given above is
f^{ -1}(x) = 3 + sqrt(x)
Matched Exercise 2: Find the inverse function of f given by
f(x) = (x + 1)^{2}, if x >= -1
Example 3: Find the inverse function of f given by
f(x) = (x + 1)/(x - 2)
Solution to example 3:
Write the function as an equation.
y = (x + 1) / (x - 2)
Multiply both sides of the above equation by x - 2 and simplify.
y (x - 2) = x + 1
Multiply and group.
y x - 2y = x + 1
y x - x = 2y + 1
Factor x on the left side and solve
x(y - 1) = 1 + 2y
x = (1 + 2y) / (y - 1)
Change x to y and y to x
y = (1 + 2x) / (x - 1)
The inverse of function f given above is
f^{ -1}(x) = (1 + 2x) / (x - 1)
Matched Exercise 3: Find the inverse function of f given by