Transcript 4-2:Composition of Functions

4-2:Composition and
Inverses of Functions
English Casbarro
Unit 4
Composition of Functions
Recall that f(5) when f(x) = 2x – 4 is evaluated by substituting the value into
the equation. f(5) = 2(5) – 4 = 10 – 4 = 6, so f(5) = 6.
The composition of functions is used when you have 2 functions. You will
still do exactly the same thing as when you have a value, but you will now
have another function. The notation for a composition is written 2
different ways:
f ( g( x ))
or
f  g( x )
.
They are both stated: “ f of g , regardless of which notation you choose.
Note: f ( g( x )) and
g( f ( x ))
are 2 different things!
Be careful not to confuse the
composition notation with the
notation for multiplication.
fg(x) ≠ f(g(x))
f
Inverse Functions
If and are functions such that
f ( g( x ))  g( f ( x )  x
then f and g are inverses of each other. The
notation is given as such: If f is the original
1
f
function, then is the inverse of f .
If f and g are inverses of
each other, then domain of f
is the same as the range of g
and vice versa.
Ex.
f ( x )  {( 31
, ),( 2,5),(5,4),( 17
, )}
The inverse of f:
Domain: {3,2, 5, –1}
Range: {1,5,4,7}
f 1( x )  {(1,3),(5,2 ),( 4,5),( 7,1)}
Notice how the x and y have changed positions. Now the domain is {1,5,4,7} and
the range is {3,2,5,–1}. [This is also the reason that the horizontal line test works]
When you are finding the inverse of a
function, you will still switch the x and the
y, but then you will solve for the y. For
example, find the inverse of f(x)=3x + 2
Step 1: Write out the function using a y instead of
function notation:
y  3x  2
Step 2: Switch the x and the y:
x  3y  2
Step 3: Solve for y:
x  3y  2
2
2
x  2  3y
x 2
y
3
or
x 2
f (x) 
3
1
(divide everything by 3)
or
x 2
f (x)  
3 3
1
When the function is not linear, you
will still follow the procedure
 Find the inverse of
f ( x )  2x  5
2
If the function is exponential, you
will need to define another term
Ex. y = 2x
In order to solve the inverse equation for y,
you will need a logarithm.
The inverse of y = 2x is log2x = y
Logarithms
Definition: A logarithm is an exponent to which a specific base has been
raised to obtain a specific value.
Exponential form
Logarithmic form
You can write an exponential equation in logarithmic form and vice versa.
Converting between exponential
and logarithmic form
Exponential
Form
26=64
Logarithmic
Form
log264=6
41=4
log44=1
5-2=0.04
Log50.04=-2
50=1
log51=0
34=81
log381=4
Solving by using mental math
 log2x=4
 log39=x
 logx27=3
 log168=¾
Turn in the following Problems
1. A theater sells tickets for $22. If you pay by credit card, the theater adds a
service charge of $3.50 to the entire order.
a. Write a function that gives the amount billed to the credit card as a function
of the number of tickets purchased.
b. Write the inverse function, and use it to find the number of tickers
purchased when the credit card bill is $157.50
c. Is it possible to have a total of $332.50 billed to your credit card for these
tickets? Why or why not?
2. For a certain credit card with 19% annual interest compounded monthly, the total
amount A that you owe after n months is given by A = P(1.016)n , where P is the
starting balance.
a. You start with a balance of $500. Write and solve a logarithmic expression for
the number of months it will take for the debt to double.
b. How many additional months will it take for the debt to double again?
c. Does the amount of time that it takes the debt to double depend
on the starting balance?