You will describe properties of inverse functions, identify when functions are invertible, and evaluate inverse functions at specified values.
After completing this tutorial, you will be able to complete the following:
Two functions f and g are inverses of each other if f(g(x)) = g(f(x)) = x. In this case, the function g is denoted by f-1 and is read as "f inverse."
Properties of inverse functions
If a function f(x) is interpreted as the mapping from set A to set B, then the function f-1(x) is the mapping from set B to set A.
If b = f(a), then a = f-1(b). Please note that the establishment of this property is a major goal of this activity object.
The inverse of f-1(x) is f(x); i.e. (f-1)-1 = f
The graphs of f(x) and f-1(x) are symmetrical with respect to the line y = x, as shown in the following graph.
The vertical line test is used test to determine whether or not a given graph represents a function; if a vertical line cuts the graph of a function at more than one point, the graph does not represent a function and the function fails this test.
Similarly, the horizontal line test is used to determine whether or not the inverse of a given graph represents a function; if a horizontal line cuts a graph at more than one point, the inverse does not represent a function.
In other words, a function must be one-to-one and onto (or "bijective") to be invertible; every element in the domain must be mapped to a different element in the range (one-to-one), and no element in the range must be left unmapped (onto).
If a function is not onto, it is not invertible. However, the same function may be onto within a certain part of its domain; in that case, it is invertible within this part of its domain. For instance, f(x) = x2 is not invertible as it is, but it becomes invertible if its domain is limited to the non-negative real numbers or the non-positive real numbers.
|Approximate Time||45 Minutes|
|Pre-requisite Concepts||Learners should be familiar with the coordinate plane, evaluating expressions, function, domain, input, output, range, and composition of functions.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||inverses of function, horizontal line test, inverse|