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Solving Quadratic Equations by Completing the Square
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Finding the Equation of an Inverse Function

Since f(x) and f -1(x) are inverses, they “undo” each other. For example, the function f(x) = x + 2 is a rule that says “add 2 to the input.” The inverse of this function is f -1(x) = x - 2. This is a rule that says “subtract 2 from the input.”

As a consequence, the composition of f(x) and f -1(x) simplifies to x.

 

Property — Composition of a Function and Its Inverse

If a function, f(x), has an inverse, f -1(x), then:

for every x in the domain of f, and

for every x in the domain of f -1.

 

Example 1

Given f(x) = 5x - 4 and , determine if g(x) is the inverse of f(x).

Solution

If g(x) is the inverse of f(x), then the composition (f g)(x) will equal x.

Find (f g)(x).

(f g)(x)

= f[g(x)]
Replace g(x) with  
In f(x), replace x with  
Cancel common factors of 5.

Subtract.

  = x + 4 - 4

= x

Since (f g)(x) = x, g(x) is the inverse of f(x).

 

Note:

We can also use (g f)(x) to see if is the inverse of f(x) = 5x - 4.

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