Math 20P Nonlinear Equations DL - M4, S1, A3 DL - M4, S1, A1-2 Operations and Composition of Functions DL - M4, S2, A1-2 Inverse of Functions DL - M4, S2, A3 Polynomial and Rational Functions DL - M4, S1, A4 DL - M4 , S3, A2 Absolute Value DL - M4, S3, A1 Prerequisite Skills

Function Inverses (Reflection in line y = x)

Discover and remember the pattern(s) by completing Help to Remember What to Learn

If you understand the pattern(s) the concept(s) in What to Learn should make sense.

What to Learn

Help to Remember What to Learn

Inverses of Functions

Note: inverse means opposite

• 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 5. 6.

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

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Started September, 1998. Copyright © 2006, 2007