Transcript 1.2 Functions and Graphs

Functions and Graphs
1.2
Symmetric about the y axis
FUNCTIONS
Symmetric about the origin
Even functions have y-axis Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
Odd functions have origin Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
A function is even if f( -x) = f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
original function back again it is even.
f x   5 x  2 x  1
4
2
Is this function even?
YES
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function even?
NO
3
3
f  x   2( x)  ( x)  2 x  x
3
A function is odd if f( -x) = - f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.
f x   5 x  2 x  1
4
2
Is this function odd?
NO
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function odd? YES
3
3
f  x   2( x)  ( x)  2 x  x
3
If a function is not even or odd we just say neither
(meaning neither even nor odd)
Determine if the following functions are even, odd or
neither.
Not the original and all
3
terms didn’t change
signs, so NEITHER.
f x   5 x  1
f  x   5 x   1  5 x  1
3
3
f x   3x  x  2
4
2
Got f(x) back so
EVEN.
f  x   3( x)  ( x)  2  3x  x  2
4
2
4
2
You should be familiar with the shapes of
these basic functions.
Library of Functions
Equations that can be
written f(x) = mx + b
slope
y-intercept
The domain of these functions
is all real numbers.
f(x) = 3
f(x) = -1
f(x) = 1
Constant Functions
f(x) = b, where b is a real number
The domain of these functions
is all real numbers.
Would constant
functions be even
or odd or neither?
The range will only be b
If you put any real
number in this function,
you get the same real
number “back”.
f(x) = x
Identity Function
f(x) = x, slope 1, y-intercept = 0
The domain of this function is
all real numbers.
Would the identity
function be even
or odd or neither?
The range is also all real numbers
Square Function
f(x) = x2
The domain of this function is
all real numbers.
Would the square
function be even
or odd or neither?
The range is all NON-NEGATIVE real numbers
Cube Function
f(x) = x3
The domain of this function is
all real numbers.
The range is all real numbers
Would the cube
function be even
or odd or neither?
Square Root Function
f x   x
The domain of this function is
NON-NEGATIVE real numbers.
Would the square
root function be
even or odd or
neither?
The range is NON-NEGATIVE real numbers
1
f x  
x
Reciprocal Function
The domain of this function is
all NON-ZERO real numbers.
Would the
reciprocal function
be even or odd or
neither?
The range is all NON-ZERO real numbers.
f x   x
Absolute Value Function
The domain of this function is
all real numbers.
Would the
absolute value
function be even
or odd or neither?
The range is all NON-NEGATIVE real numbers
Recall: These are functions that
are defined differently on different
parts of the domain.
 x, x  0
f x    2
x , x  0
This means for x’s less than
0, put them in f(x) = -x but for
x’s greater than or equal to
0, put them in f(x) = x2
What
Whatdoes
doesthe
thegraph
graph
of
off(x)
f(x)==-x
x2look
looklike?
like?
Remember yy==f(x)
f(x)so
solets
let’s
Remember
graphyy==x-2xwhich
whichisisaasquare
line of
graph
slope –1(parabola)
and y-intercept 0.
function
Since
Since we
we are
areonly
only
supposed
supposed to
tograph
graphthis
thisfor
for
xx<
 0, we’ll only
stop keep
the graph
the
right
at x =half
0. of the graph.
This then is the graph
for the piecewise
function given above.
2 x  5,  3  x  0

f x    3,
x0
 5 x,
x0

For
Forxx>=00the
thefunction
function
For x values between
isvalue
to be
–3supposed
andis0supposed
graph
the to
along
be
so2x
line
plot
ythe
= - 5x.
line–3
ythe
=
+ 5.
point (0, -3)
Since you know this
the graph
graph is
is aa
piece of a line, you can just plug in
each
0
to see
endwhere
value to
to start
get the
the line and
endpoints.
then
count af(-3)
– 5 =slope.
-1 and f(0) = 5
So this the graph of
the piecewise
function
solid dot for
"or equal to"
open dot
since not
"or equal to"
You try one:
Graph the function described by:
3x x  3
h( x )   2
x
x

3

f  g ( x)  f ( g ( x))
“f of g of x”
f ( x)  x  7
g ( x)  x 2
2
f ( g ( x))  x  7
f (g (2))   3