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Composite and Inverse Functions , Lecture notes of Algebra

Composite function is a merge of multiple functions in simple form.

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2021/2022

Uploaded on 02/03/2022

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8.4 Composite and Inverse Functions
The Composition of Functions
The composition of the function f with g is denoted by
fg
and is
defined by the equation
fg
( )
x
( )
=f g x
( )
( )
The domain of the composite function
fg
is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.
Forming Composite Functions
Example 1: Given f(x) = 5x + 2 and g(x) = 3x 4, find (
fg
)(x)
and (
gf
)(x).
Inverse Functions
Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g, and
g(f(x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and is denoted by
f1
(read “f-inverse”). Thus f(
f1
(x))= x and
f1
(f(x)) = x. The domain of f
is equal to the range of
f1
and vice versa.
Verifying Inverse Functions
Example 2: Verify that each function is the inverse of the other:
f(x) = 6x and g(x) =
x
6
pf3
pf4
pf5

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8.4 Composite and Inverse Functions

The Composition of Functions The composition of the function f with g is denoted by f  g and is defined by the equation

( f^ ^ g) ( x) =^ f^ (g (^ x ))

The domain of the composite function f  g is the set of all x such that

  1. x is in the domain of g and
  2. g(x) is in the domain of f. Forming Composite Functions Example 1: Given f(x) = 5x + 2 and g(x) = 3x − 4, find ( f  g)(x) and ( g  f)(x). Inverse Functions Let f and g be two functions such that f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f. The function g is the inverse of the function f, and is denoted by f − 1 (read “f-inverse”). Thus f( f − 1 (x))= x and f − 1 (f(x)) = x. The domain of f is equal to the range of f − 1 and vice versa. Verifying Inverse Functions Example 2: Verify that each function is the inverse of the other: f(x) = 6x and g(x) = x 6

Example 3: Verify that each function is the inverse of the other. f (^) ( x) = 4 x + 9 and g (^) (x ) = x − 9 4 Finding the Inverse of a Function The equation for the inverse of a function can be found as follows:

  1. Replace f(x) with y in the equation for f(x).
  2. Interchange x and y.
  3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
  4. If f has an inverse function, replace y in step 3 with f − 1 (x). We can verify our result by showing that f( f − 1 (x))= x and f − 1 (f(x)) = x. Example 4: Find the inverse of f(x) = 6x + 3

b. f (^) (x ) = x 2

  • 2 x + 1 c. f(x) = x 3 − 1 d. f (^) ( x) = x

e. f (^) ( x) = 2 x + 1 Graphs of f and f1 The graphs of f and f − 1 are reflections of one another through the line y = x. Points on the graph of f − 1 can be found by reversing the coordinates of the points on the graph of f. Example 8: Consider the graph of the function f traced by joining the points given below with straight-line segments. Sketch the graph of f and the graph of f − 1 . Points on y = f(x) (^) Points on y = f − 1 (x) (−2,0) (0,1) (1,3)