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ZingPath: Composition and Inverses

Fundamental Concepts on the Inverses of Functions

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Composition and Inverses

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Fundamental Concepts on the Inverses of Functions


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You will describe properties of inverse functions, identify when functions are invertible, and evaluate inverse functions at specified values.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Describe the inverse of a function.
  • Explain the properties of inverse functions.
  • Identify if a function is invertible given its Venn diagram.
  • Identify if a function is invertible given its graph.
  • Evaluate a function or its inverse at a specified value given its equation, table, or Venn diagram.

Everything You'll Have Covered

Inverse functions

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

Property 1:

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.

Property 2:

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.

Property 3:

The inverse of f-1(x) is f(x); i.e. (f-1)-1 = f

Property 4:

The graphs of f(x) and f-1(x) are symmetrical with respect to the line y = x, as shown in the following graph.

Property 5:

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.

Tutorial Details

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.
Course Algebra-1
Type of Tutorial Concept Development
Key Vocabulary inverses of function, horizontal line test, inverse