Questions on
inverse functions are presented along with detailed solutions and explanations. The questions below will help you develop the computational skills needed in solving questions about inverse functions and also gain deep understanding of the concept of inverse functions.

Question 1:

Find the parameters a and b included in the linear function f(x) = a x + b so that f^{ -1} (2) = 3 and f^{ -1} (-3) = 6, where f^{ -1} (x) is the inverse of function f.

Solution to Question 1:

From the properties of inverse functions if f^{ -1} (2) = 3 and f^{ -1} (-3) = 6, then

f(3) = 2 and f(6) = - 3

Use the above to write

f(3) = 3a + b = 2 and f(6) = 6a + b = -3

Solve the 2 by 2 system of equations 3a + b = 2 and 6a + b = -3 to obtain

a = - 5 / 3 and b = 7

Question 2:

Given f(x) = x^{3} + 2 x, complete the table of values given below and find f ^{-1}(3) and f ^{-1}(- 12).

x

f(x)

0

0

1

3

2

12

Solution to Question 2:

Since f(1) = 3, from the properties of the inverse functions, we have

f ^{-1}(3) = 1

f is odd function and therefore f(- 2) = - f(2) = -12. Hence

f ^{-1}(- 12) = - 2

Question 3: Proove that the inverse of an invertible odd function is also an odd function.

Solution to Question 3:

Start with the property of f and its inverse f^{ -1}

f ( f^{ -1}(x)) = x

Change the right side x of the above equation to - (-x) and write

f ( f^{ -1}(x)) = - ( - x)

Again change - x in the above equation to f ( f^{ -1}( - x)) and write

f ( f^{ -1}(x)) = - f ( f^{ -1}( - x))

Since f is odd, the right side in the above equation may written as follows

- f ( f^{ -1}( - x)) = f( - f^{ -1}( - x) )

Hence

f ( f^{ -1}(x)) = f( - f^{ -1}( - x) )

Which gives

f^{ -1}(x) = - f^{ -1}( - x)

and prooves that f^{ -1} is also odd.

Question 4:

Let f(x) = 1 / (x - 2). Find the points of intersection of the graphs of f and that of f^{ -1} the inverse of function f. Graph f, its inverse and the line y = x. Where are the points of intersection located?

Solution to Question 4:

We first find the formula for f^{ -1}(x)

y = 1 / (x - 2)

Change y into x and x into y.

x = 1 / (y - 2)

Solve the above for y.

y = 1 / x + 2 = f^{ -1}(x)

To find the points of intersection of the graphs of f and f^{ -1}, we need to solve for x the equation

f(x) = f^{ -1}(x)

1 / (x - 2) = 1 / x + 2

The above equation has the solutions

x = 1 + √2 and x = 1 - + √2

The y coordinate is given by

for x = 1 + √2 , y = f(1 + √2) = 1 + √2

for x = 1 - √2 , y = f(1 - √2) = 1 - √2

The points of intersections are given by their x and y coordinates as follows

(1 + √2 , 1 + √2) and (1 - √2 , 1 - √2)

The graph of f (in red) and that of f^{ -1} (in blue) are plotted below, along with y = x and the points of intersection which are located on the line y = x.

Question 5:

Graph function f defined by f(x) = |x - 2| + 2x and its inverse f ^{-1} and find a formula for its inverse.

Solution to Question 5:

For (x - 2) < 0, |x - 2| = - (x - 2) and f(x) is given by(x)

f(x) = - (x - 2) + 2x = x + 2

For x - 2 ≥ 0, |x - 2| = (x - 2) f(x) is given by(x)

f(x) = x - 2 + 2x = 3x - 2

The graph of f (in black) is made up of two linear parts. In order to graph the inverse of , we need to determine points (a,b) on the graph of and then use them to graph the inverse. The points are

(-2 , 0) , (2 , 4) , (3 , 7)

On the graph of the inverse function, these points become

(0 , -2) , (4 , 2) , (7 , 3)

Graph them and complete the graph of the inverse function (in blue) using reflection on the line y = x. The graph of the inverse is also made up of two linear parts as shown in the figure below.

If we examine the formula of f and its graph we can assume that the inverse function has a formula of the form:

f ^{-1}(x) = a|x - 4| + bx + c

Coefficient a, b and c are determined using the points (0 , -2) , (4 , 2) , (7 , 3) on the graph of f ^{-1} as follows:

f ^{-1}(0) = a|0 - 4| + b(0) + c = - 2

f ^{-1}(4) = a|4 - 4| + b(4) + c = 2

f ^{-1}(x) = a|7 - 4| + b(7) + c = 3

We now solve the system of the above equations with unknown a, b and c to find the values:

a = - 1 / 3 , b = 2 / 3, c = - 2 / 3

Which gives f ^{-1}(x) = (-1/3)|x - 4| + 2 x / 3 - 2 / 3