Transcript Function Composition

Function
Compositions
and
Inverses
Function Notation
• f(x) does NOT stand for MULTIPLICATION!
• We read f(x) “f of x” and it means that the
function’s independent variable is x.
• If the function is defined as f(x) = 3x - 1
and we are asked to find f(2), we just
substitute 2 for x in the function:
f(2) = 3(2) - 1 = 6 - 1 = 5
Function Substitutions
2
x
Given g(x) = - x, find g(-3)
Substitute -3 for x:
2
g(-3) = (-3) - (-3)
= 12
g(-3) = 12
Function Substitutions
2
4x
Given g(x) = 3x + 2, find g(5)
Substitute 5 for x
2
g(5) = 3(5) - 4(5) + 2
= -83
g(5) = -83
Function Composition
Function Composition is just more
substitution, very similar to what we
have been doing with finding the
value of a function. The difference
is we will be substituting another
function instead of a number ...
Function Composition
For example…
Given f(x) = x - 5, find f(a+1)
Substitute (a+1) for x
f(a + 1) = (a + 1) - 5
= a+1 - 5
The answer is a
function in terms
of ‘a’
=a-4
Function Composition
• Composition notation looks like
g(f(x)) or f(g(x)), we read this
‘g of f of x’ or ‘f of g of x’.
• We are given f(x) and g(x), the
function inside the parentheses gets
substituted into the other.
Function Composition
Given the functions:
f(x) = 2x+2 & g(x) = 2
find f(g(x))
This notation tells us to substitute the g(x)
function, 2, for x in the f(x) function:
f (2) = 2(2)+2
=6
Function Composition
Given the functions:
g(x) = x - 5 & f(x) = x + 1
find f(g(x))
This notation tells us to substitute the g(x)
function, x-5, for x in the f(x) function:
f (x-5) = (x-5)+1
=x-4
Function Composition
Reverse the composition:
g(x) = x - 5 & f(x) = x + 1
find g(f(x))
This notation tells us to substitute the f(x)
function, x+1, for x in the g(x) function:
g(x+1) = (x+1)-5
=x-4
Function Composition
In the last example, f(g(x)) and g(f(x))
had the same results. This is not always
the case.
Try this example:
f(x) = x2 + x & g(x) = x - 4
find f(g(x)) and g(f(x))
Function Composition
f(x) =
2
x
+ x & g(x) = x - 4
1) f(g(x)) = f(x-4) = (x-4)2 + (x-4)
= x2 -8x+16+x-4
= x2 -7x+12
2) g(f(x)) = g(x2 + x ) = (x2 + x )-4
= x2 + x - 4
Function Composition
New Example:
Given f(x) = 2x + 5 & g(x) = 8 + x
find f(g(-5) & g(f(-5)
1) f(g(-5) : find g(-5) = 8 + (-5) = 3
then find f(3) = 2(3) + 5 = 11
2) g(f(-5)) : find f(-5) = 2(-5) + 5 = -5
then find g(-5) = 8 + (-5) = 3
Function Inverse
Remember: a function is a set of ordered
pairs (including lists of discrete points and
also equations which give us infinite points),
where no two points have the same xcoordinate.
The Inverse of a function is the set of points
where each point in the function is reversed,
(y, x).
Function Inverse
A function that is a list of ordered pairs is
easy to find the inverse of:
f(x) = {(1, 2), (2, 5), (3, -4), (4, 0)}
The inverse is:
f-1(x) = {(2, 1), (5, 2), (-4, 3), (0, 4)}
Function Inverse
To find the inverse of a function that is
written as an equation, like:
f(x) = x + 7
We will:
1) Replace the function label, f(x) with y
2) Swap the variables, x and y
3) Solve the new equation for y
Function Inverse
Find the Inverse of:
f(x) = x + 7
Replace:
Swap:
Solve:
y=x+7
x=y+7
y=x-7
f-1(x) = x - 7
Function Inverse
Find the Inverse of:
f(x) = 3x - 4
Replace:
Swap:
Solve:
y = 3x - 4
x = 3y - 4
3y = x + 4
y = (x + 4)/3
f-1(x) = (x + 4)/3
Function Inverse
Find the Inverse of:
f(x) = (2x + 5)/3
Replace:
Swap:
Solve:
y = (2x + 5)/3
x = (2y + 5)/3
3x = 2y + 5
2y = 3x - 5
f-1(x) = (3x - 5)/2
Function Inverse
Find the Inverse of:
f(x) = x2 - 4
Replace:
Swap:
Solve:
y = x2 - 4
x = y2 - 4
y2 = x + 4
y = ±√x + 4
Function Inverse
Note: In the last example, the inverse does
NOT pass the test to be a function. This
sometimes happens, that the inverse of a
function is not a function.
This occurs when the function has points
with the same y-value (allowed in
functions).
Function Inverse
A function whose inverse IS ALSO a
function is called a ONE-TO-ONE function.
Each x-coordinate has a different ycoordinate and each y-coordinate has a
different x-coordinate.