Download 1.5 Combinations of Functions

Domain and Range
Domain
• The domain is the set of all x values that work
in the function
• To find the domain:
– What kind of numbers can I plug in for x??
• Positive numbers??
• Negative numbers??
• Zero??
Range
• The range is the set of all y values that a
function outputs
• To find the range:
– you have to think!! what numbers can you get out
of the function??
Polynomials
• f(x) = -2x4 – 6x2 + 5
• g(x) = 2x + 1
• y = (x – 9)2
D: All Real Numbers
Absolute Value
• f(x) = |x – 2|
• g(x) = 3|2x – 1|
• y = -4 |x + 1| + 3
Radical Functions
f (x) = 4x + 3
g(x) = -x + 2 + 5
y=- x
Rational Functions
2x
f (x) =
x+ 4
3
g(x) = 2
x +1
5
y= 2
x - 25
1.5 Combinations of Functions
Arithmetic Combinations of Functions
• Sum: (f + g)(x) = f(x) + g(x)
• Difference: (f – g)(x) = f(x) – g(x)
• Product: (fg)(x) = f(x) · g(x)
• Quotient:
f
f ( x)
 ( x) 
, g ( x)  0
g ( x)
g
Find the sum, difference, product, and
quotient of the two functions.
1. f(x) = 2x + 1
g(x) = x2 + 2x – 1
2. f(x) = x2
g(x) = x – 3
3. f ( x)  x
g ( x)  4  x 2
Compositions of Functions
The composition of the function f with the
function g is:
 f  g ( x) 
f ( g ( x))
The domain of f ᵒ g is the set of all x in the
domain of g such that g(x) is in the domain of f
Finding compositions of functions and
their domains
Find  f  g  and g  f  then their domains
1.
f ( x)  x
2.
f ( x)  9  x
g ( x)  x  1
g ( x)  x 2  9
Identifying a Composite function
Write h(x) as a composition of two functions
1. h(x) = (3x – 5)3
2.
1
h( x ) 
3x  1
Bacteria Count
The number N of bacteria in a refrigerated food is given
by N (T )  20T 2  80T  500,2  T  14
where T is the temperature of the food in degrees Celsius.
When the food is removed from the refrigeration, the
temperature of the food is given by
T (t )  4t  2,0  t  3 where t is time in hours.
a. Find the composition N(T(t)) and interpret its meaning
b. Find the number of bacteria in the food when t = 2
hours
c. Find the time when the bacterial count reaches 2000.
Combinations of functions using a
graph