Inverse of Functions

2^-1 means multiplicative inverse ( reciprocal) of 2 = 1/2.
f^-1(x) means inverse (exchange domain/range) of the function f(x).
- For f(x): If f(x) = 2x + 3, then y = 2x + 3
- But for f^-1(x): x = 2y + 3 y = (x - 3)/2

For Pure Math 20 focus on the observation that the function and its inverse are reflections in the line y = x.
Vertical line test: a vertical line cannot intersect the curve of a function in more than 1 place. The inverses of functions 2 and 4 do not pass the vertical line test.

Interactive Activity

Click on the applet to activate
Click one of the 6 function icons.
Observe the grey mirror line is y = x.
Study the relationship of the corresponding points on the graphs of the function and its inverse.
Move the parameter sliders for a, b, h and k. These parameters will be studied in detail in Pure Math 30.
Select SET FUNTION to explore the effect of the a, b, h and k parameters on other functions.

1. y = a[b(x-h)]+k

x = a[b(y - h)]+k

2. y=a[b(x-h)] ²+k

x = a[b(y-h)] ²+k

3. y = a[b(x-h)] ³+k

x = a[b(y-h)] ³+k

4. y = a|b(x-h)|+k

x = a|b(y-h)|+k

Linked Source - Ron Blond

Example

Graph f(x) = x² + 3 and its inverse.

Solution
Replace f(x) with y for ease in finding the inverse.

y = x² + 3

Find the inverse.

Note: The graphs of a an original function and it's inverse are always reflections in the line y = x.

Notice from the graph that the relation x = y² + 3 is not a function, since y is not uniquely determined for x > 3. In other words there are more than one y-value for each x-value in the region x > 3.

The vertical line test is the test for functions. Imagine a vertical line moving across the grid. If the line never touches the relation at more than one point, then the relation is a function.

If you were to restrict the original function to y = x² + 3, where x > 0, then its inverse would be x = y² + 3, where y > 0. The negative portions of each graph would be eliminated. As a result, the inverse could be rewritten as or .

Recall that the inverse will undo what the function does. For example, the function takes 4 into 19; that is, f(4) = 19. The inverse will take 19 back into 4; that is, f^-1(19) = 4.

There are many ways of restricting the domain of a function so that its inverse is also a function. In the previous example, you could have written f(x) = x² + 3, x < 0. The inverse would be x = y² + 3 + 3, y < 0. When you solve for y, the inverse is or .

Other Examples:

The domain of f(x) is the set of reals,and the range is f(x) > 1.

The domain of f^-1(x) is x > 1and the range is the set of reals,.

Inverse Check

Parts of this work has been adapted from a Math 20 Pure learning resource originally produced and owned by Alberta Education

Comments to: Jim Reed - Homepage