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First part is out of 70 pts.
Second part is out of 35 pts.
Total points possible is 105 pts.
 Grade is out of 100 pts.
Transformations
We're going to take the functions from
last class and alter them to get new ones
Ex. Compare y = x2 and y = x2 + 2
6
gx = x2+2
4
f x = x2
2
Ex. Compare y = x2 and y = (x + 2)2
6
4
f x = x2
2
gx = x+22
-5
Ex. Use the graph of f (x) = x3 to sketch:
a. f (x) = x3 – 1
b. f (x) = (x + 2)3 + 1
2
2
-2
-2
Ex. Given the graph of y = x 4 below, identify
the equation of the second graph.
4
2
f x = x4
2
-2
xxbelow to
Ex. Use the graph of f
describe and draw the graph of:
a) f
x
x
g  x =
2
2
-2
b) fx


x

2

x
2
-2
Multiplying by a number will cause a
stretch or shrink
This is called a nonrigid transformation
You could memorize what each does, but
it's easier to figure it out by plugging in
numbers.
x
xbelow to
Ex. Use the graph of f
draw the graph of:
2
a) f
x
3
x
2
-2
-2
1
f
x

x

b)
3
2
-2
Practice Problems
Section 2.5
Problems 1, 9, 19, 27
Composite Functions
This means combining two functions to get
a new function.
Ex. Let f (x) = 2x – 3 and g(x) = x2 – 1, find
a) ( f + g)(x)
b) ( f g)(x)
f
c)   x
g
Ex. Let f (x) = 2x + 1 and g(x) = x2 + 2x – 1,
find ( f – g)(2).
fgx
 
Ex. Given f (x) = x + 2 and g(x) = 4 – x2
f gx
means

a) f gx
gf
x

b) 
Ex. Write hx
 
1
x2
of two functions.
2
as the composition
Practice Problems
Section 2.6
Problems 13, 31, 35, 47
Inverse Functions
Ex. Make t-charts for f (x) = x + 4 and
g(x) = x – 4.
Two functions are inverses if the roles of x
and y are switched.
3
fx

Ex. If  x1
, find f -1(x).
An inverse function doesn't always exist, and
you won't always be able to solve for y
Ex. For the previous example, find
( f -1∘ f )(x).
For any function, ( f -1∘ f )(x) = x
5
5
Ex. Show that fx
and gx
  2
x

2
x
are inverses.
Because the x 's and y 's are switched, the
graphs of f and f -1 are reflected over the
line y = x.
Ex. Given the graph of f (x) below, sketch f -1.
2
-2
f(x)
y x and y = x2 are not inverses.
- Consider the graphs
2
-2
A function is one-to-one if each y-coordinate
corresponds to exactly one x-coordinate
2
-2
f(x)
These graphs pass the horizontal line test
2
f(x)
-2
A function is invertible (it has an inverse) if
it passes the horizontal line test
Ex. Are these functions invertible?
a)
2
-2
b)
2
-2
Even if the function is invertible, you still
may not be able to find an equation.
Practice Problems
Section 2.7
Problems 15, 17, 29, 55