Combinations of Functions &
Inverse Functions
Obj: Be able to work with combinations/compositions of
functions. Be able to find inverse functions.
TS: Making decisions after reflections and review
Warm-Up:
Given f(x) = 2x + 3 and g(x) = x2 – 4, find…
f(2) + g(2)
g(3) – f(3)
Arithmetic Combinations
( f  g )  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
 f  f ( x)
 
 g  g ( x)
( fg )( x)  f ( x)  g ( x)
Where the domain is the real numbers that
both f and g’s domains have in common.
For f/g also g(x) ≠ 0.
Examples
Find each of the below combinations given
f ( x)  x
1) (f∙g)(2)
2) (f – g)(2)
g ( x)  4  x 2
3) (f/g)(x) & its domain
Arithmetic Compositions
(f○g)(x) = f(g(x))
Given the below find each of the following.
1
f ( x) 
x
1) (f ○ g)(2)
1
g ( x)  2
x  2x  3
2) g(f(-1))
Arithmetic Compositions
(f○g)(x) = f(g(x))
Given the below find each of the following.
1
f ( x) 
x
1
g ( x)  2
x  2x  3
3) f(g(x)) & its domain
Find f(g(x)) and g(f(x))
f ( x)  2 x  3
1
g ( x)  ( x  3)
2
Definition of the Inverse of a
Function
Let f and g be two functions where 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. Under these
conditions, g is the inverse of f and g is denoted
f-1.
Thus f(f-1(x))=x and f-1(f(x))=x where the
domain of f must equal the range of f-1 and
the range of f must equal the domain of f-1.
Graphs of Inverses
Two equations are inverses if their graphs are
reflections of one another across the line y=x.
y
f
y=x
f -1
1
x
1
Inverse FUNCTIONS
A function f(x) has an inverse function if the graph of f(x)
passes the ___________________.
(In other words the relation is ONE-TO-ONE: For each y
there is exactly one x)
Circle the functions that are one-to-one (aka have inverse
functions)
y
y
x
y
x
y
x
y
x
x
Examples:
Determine if the two functions f and g are
inverses.
1)
f ( x)  3 x  2
and
1
g ( x)  x  2
3
Finding an Inverse
1)Verify that the function is one-to-one thus has an
inverse function using the Horizontal Line Test.
2)Switch x & y.
3)Solve for y.
4)Make sure to use proper inverse notation for y for
your final answer. (Ex: f-1(x), not y)
To check your answer:
Verify they are inverses by testing to see if
f(f-1(x)) = f-1(f(x)) = x
Find the inverse function if there is one, if there is not one,
restrict the domain to make it one-to-one then find the
inverse function .
1) f(x) = -.5x + 3
Find the inverse function if there is one, if there is not one,
restrict the domain to make it one-to-one then find the
inverse function .
2) f(x) = x2 – 4