Math 030 - Cooley
Intermediate Algebra
OCC
Section 9.1 – Composite Functions and Inverse Functions
Composition of Functions
The composite function f g , the composition of f and g, is defined as
f
g  x   f  g  x  
One-To-One Function
A function f is one-to-one if different inputs have different outputs. That is, f is one-to-one if for any a and b in
the domain of f with a ≠ b, we have f (a)  f (b) . If a function is one-to-one, then its inverse correspondence is
also a function.
The Horizontal Line Test
If it is impossible to draw a horizontal line that intersects a function’s graph more than once, then the function
is one-to-one. For every one-to-one function, an inverse function exists.
To Find A Formula For f-1 (or Find The Inverse Of f)
First make sure that f is one-to-one.
1) Replace f ( x) with y.
2) Interchange x and y. (This gives the inverse function.)
3) Solve for y.
4) Replace y with f 1 ( x) . (This is the inverse function notation.)
Visualizing Inverses
The graph of f 1 is a reflection of the graph of f across the line y  x
Composition And Inverse
If a function f is one-to-one, then f 1 is the unique function for which
( f 1 f )  x   f 1 ( f ( x))  x
and
(f
f 1 )  x   f ( f 1( x))  x .
 Exercises:
For each pair of functions, find (a) ( f g )(1) ; (b) ( g f )(1) ; (c) ( f g )( x) ; (d) ( g f )( x) .
1)
f ( x)  x  4; g ( x)  x 2  5
2)
f ( x)  10  x; g ( x)  x
3)
f ( x)  x 2  8; g ( x)  x  17
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Math 030 - Cooley
Intermediate Algebra
OCC
Section 9.1 – Composite Functions and Inverse Functions
 Exercises:
Determine whether each function is one-to-one.
4)
f ( x)  x  5
5)
7
6)
y
f ( x)  3  x 2
7
7)
6
5
5
4
4
3
3
2
2
1
–7 –6 –5 –4 –3 –2 –1
–1
1
x
1
2
3
4
5
6
y
6
–7 –6 –5 –4 –3 –2 –1
–1
7
–2
–2
–3
–3
–4
–4
–5
–5
–6
–6
–7
–7
x
1
2
3
4
5
6
7
For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
8)
f ( x)  x  2
10)
f ( x) 
12)
f ( x)  x  1
4
x
9)
f ( x)  x 2  4
11)
f ( x)  ( x  1)3
13)
f ( x)  x 2  1, x  0
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Math 030 - Cooley
Intermediate Algebra
OCC
Section 9.1 – Composite Functions and Inverse Functions
 Exercises:
Solve.
14)
In exercise #11, graph the function and
its inverse using the same set of axes.
7
y
6
5
4
3
2
1
x
–7 –6 –5 –4 –3 –2 –1
–1
1
2
3
4
5
6
7
–2
–3
–4
–5
–6
–7
15)
In exercise #12, graph the function and
its inverse using the same set of axes.
7
y
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1
–1
x
1
2
3
4
5
6
7
–2
–3
–4
–5
–6
–7
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