precalculus inverse relations and functions. if two relations or functions are inverses, one...
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Inverse Relations and Functions
If two relations or functions are inverses, one relation
contains the point (x, y) and the other relation contains the point (y, x).
*Their graphs will be symmetric with respect to
the line y = x.
One-to-One / Horizontal Line Test
A function is one-to-one if no two x values have the same y value.
We can determine if a function is one-to-one by evaluating the graph of the original function using the Horizontal Line Test.
If a horizontal line passes through the graph in more than one point, the function is not
one-to-one, and the inverse of the function is not a function.
Definition of an Inverse Function
Let f and g be two functions such that the following two conditions are met.
1. f(g(x))= x for every x in the domain of g
2. g(f(x))= x for every x in the domain of f
If these two conditions are met, then g(x) is the inverse of f(x) and is denoted .
Verify that f(x) and g(x) are inverse functions algebraically.
Verify that f(x) and g(x) are inverse functions algebraically.
g
Which of the following statements is true?
A. The inverse of is .B. The function f(x) = 5 is one-to-one.C. If , then .D. The domain of f is the same as the range of
Extra Practice
Write the equation of a function, g, such that the inverse of g will have the domain .