Download 7.3 Power Functions & Function Operations

7.3 Power Functions &
Function Operations
p. 415
Sum :
f(x) + g(x) = (f+g)(x)
Difference :
f(x) - g(x) = (f-g)(x)
Product :
f(x) * g(x) = (fg)(x)
Quotient :
f ( x)  f 
  ( x) , g ( x)  0
g ( x)  g 
f(x) = 2x – 3
Sum
Difference
Product
Quotient
and
g(x) =
x 1
2
f ( x)  g ( x)  (2 x  3)  ( x  1)
2
 x  2x  4
2
f ( x)  g ( x)  (2 x  3)  ( x 2  1)
2
 x  2x  2
f ( x) * g ( x)  (2 x  3) * ( x  1)
2
 2 x 3  3x 2  2 x  3
f ( x) (2 x  3)
 2
g ( x) ( x  1)
, x  1
In order for a relationship to
be a function…
Functions
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN
HAVE THE SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE OUTPUT
INPUT
(DOMAIN)
FUNCTION
MACHINE
OUTPUT
(RANGE)
Look on page 67
• No two ordered pairs can have
the same first coordinate
(and different second coordinates).
Time of Day
1
2
Degrees C
4
9
3
15
5
3
14
4
7
13
10
6
11
12
6
1
5
2
8
Domain
Contains the Range
Inputs: 1,2,3,4,5,6
Outputs: 9,10,12,13,15
(1,9), (2,12), (3,13), (4,15), (5,12), (6,10)
Ex.
{(2,5) , (3,8) , (4,6) , (7, 20)}
{(1,4) , (1,5) , (2,3) , (9, 28)}
{(1,0) , (4,0) , (9,0) , (21, 0)}
Notation
f (x )
“f of x”
Input = x
Output = f(x) = y
Ex: Let f(x)=3x1/3 & g(x)=2x1/3. Find (a)
the sum, (b) the difference, and (c) the
domain for each.
(a) 3x1/3 + 2x1/3
= 5x1/3
(b) 3x1/3 – 2x1/3
= x1/3
(c) Domain of (a) all real numbers
Domain of (b) all real numbers
Ex: Let f(x)=4x1/3 & g(x)=x1/2. Find (a)
the product, (b) the quotient, and (c) the
domain for each.
5
(a) 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6
(b)
4x
x
1
3
 x
6
(c) Domain of (a) all reals ≥ 0,
because you can’t take the 6th root of
a negative number.
1
2
= 4x1/3-1/2
= 4x-1/6
= 4
x
4
Domain of (b) all reals > 0,
because you can’t take the 6th root of
a negative number and you can’t have
4 a denominator of zero.
1
6
6
x
Evaluate (f-g)(x) when x = 2 for the functions
g ( x)  x  2 x  1
f ( x)  2 x  1
(f - g)(x) =
2
(2 x  1)  ( x  2 x  1)
2
2x 1  x  2x 1
2
 x2  2
(f - g)(2) =
 22  2
 2
Composition
• f(g(x)) means you take the function g and
plug it in for the x-values in the function f,
then simplify.
• g(f(x)) means you take the function f and
plug it in for the x-values in the function g,
then simplify.
The COMPOSITION of the function f with g is
( f  g )( x)  f ( g ( x))
f ( x)  x 2
g ( x)  x  1
( f  g )( x)  f ( g ( x))  f ( x  1)  ( x  1)
( g  f )( x)  g ( f ( x))  g ( x )  x  1
2
2
Plug the second function into the first
2
Evaluate the following when x = 0, 1, 2, 3 given that
f ( x)  x  2
g ( x)  4  x 2
( f  g )( x)  f (4  x 2 )  (4  x 2 )  2
( f  g )(0) 
2
0 6
6
( f  g )(1) 
 12  6
5
( f  g )( 2) 
 22  6
2
( f  g )(3) 
3 6
2
 3
 x2  6
Ex: Let f(x)=2x-1 & g(x)=x2-1. Find (a)
f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the
domain of each.
(a) 2(x2-1)-1 =
(b) (2x-1)2-1
= 22x-2-1
4
= 2 1
x
2
x2 1
(c) 2(2x-1)-1
= 2(2-1x)
= 2x  x
2
(d) Domain of (a) all reals except
x=±1.
Domain of (b) all reals except x=0.
Domain of (c) all reals except x=0,
because 2x-1 can’t have x=0.
Journal
• When I hear someone say
“Math is Fun” I…
– 5 sentences minimum
Assignment