Composite and Inverse Functions
Chapter 9 Section 2
Composite and Inverse Functions
Page 676
Warm up
Composite Functions
Rewrite two functions as one.
Notation: composite function
the composition of the function f with g.
This is read, f of g of x
x is in the domain of g
g(x) is in the domain of f
Example
f(x) = x – 300, g(x) = 0.85x find
Solution:
Since is the same as f(g(x))
Replace g(x) with 0.85x
f(0.85x)
So f(0.85x) = 0.85x – 300
Then write the answer: =0.85x - 300
Form a Composite Function
Given: f(x) = 3x – 4 and g(x) =
Find and g(f(x))
One More
Inverse Functions
Notation:
Definition: Let ‘f ‘and ‘g’ be two functions such that
f(g(x)) = x for every x in the domain of g
g(f(x)) = x for every x in the domain of f
Function ‘g’ is the inverse of the function ‘f’
read ’f inverse’.
Verify Inverse Functions
f(x) = 5x and g(x) =
To verify that f(x) and g(x) are inverses, show that
f(g(x) = x and g(f(x)) = x
Find the Inverse of a Function
1)
Replace f(x) with ‘y’ in the equation.
2)
Interchange ‘x’ and ‘y’
3)
Solve for ‘y’
If the function does not have an inverse, stop.
4) If ‘f’ has an inverse, then replace ‘y’ with
Find the inverse
Horizontal Line Test and One-to—One
Functions
A function ‘f’ has an inverse that is a function, , if there is no
horizontal line that interests the graph of the function ‘f’ at more than
one point.
If the function passes the horizontal line test, the function is call a one-
to-one function.
Which graph passes the Horizontal Line Test?
a, b, c, d ?
Which graphs represent functions that have
inverse function?
Explain how you know.
Summary
•
Composite functions.
•
Inverse functions.
•
Horizontal line test.
Learn about composite functions, inverse functions, and how to find their compositions and inverses through examples and step-by-step explanations. Explore the concept of forming composite functions, verifying inverse functions, and finding the inverse of a function using interchange and solving methods. Discover the horizontal line test and one-to-one functions to deepen your understanding of function operations.
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Presentation Transcript
Chapter 9 Section 2 Composite and Inverse Functions Page 676
Warm up 1) If f(x) = 3x 2 find f(4). 2) If g(x) = 2?2+ 7? 6 , find g(3)
Composite Functions Rewrite two functions as one. Notation: composite function ( ) f g x ( ) the composition of the function f with g. This is read, f of g of x x is in the domain of g g(x) is in the domain of f
Example f(x) = x 300, g(x) = 0.85x find Solution: Since is the same as f(g(x)) Replace g(x) with 0.85x f(0.85x) So f(0.85x) = 0.85x 300 Then write the answer: =0.85x - 300
Form a Composite Function x2+6 Given: f(x) = 3x 4 and g(x) = Find and g(f(x))
One More Find (? ?) (x) if f(x) = 2x and g(x) = x + 7
Inverse Functions f-1 Notation: Definition: Let f and g be two functions such that f(g(x)) = x for every x in the domain of g g(f(x)) = x for every x in the domain of f Function g is the inverse of the function f read f inverse . f-1
Verify Inverse Functions x 5 f(x) = 5x and g(x) = To verify that f(x) and g(x) are inverses, show that f(g(x) = x and g(f(x)) = x
Find the Inverse of a Function 1) Replace f(x) with y in the equation. 2) Interchange x and y 3) Solve for y If the function does not have an inverse, stop. 4) If f has an inverse, then replace y with f-1x ( )
Find the inverse a) f(x) = 2x + 3 b) f(x) = ?3 1
Horizontal Line Test and One-toOne Functions f-1 A function f has an inverse that is a function, , if there is no horizontal line that interests the graph of the function f at more than one point. If the function passes the horizontal line test, the function is call a one- to-one function.
Which graph passes the Horizontal Line Test? a, b, c, d ?
Which graphs represent functions that have inverse function? Explain how you know.
Summary Composite functions. Inverse functions. Horizontal line test.
