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Title: PRE CALCULUS FUNCTION NOTES
Description: pre cal function notes teaches you how to add multiply divide and subtract functions (FoG)
Description: pre cal function notes teaches you how to add multiply divide and subtract functions (FoG)
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1
...
Then for all values of x in the
intersection, the algebraic combinations of f and g are defined by the following rules:
Sum:
Difference:
Product:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(fg)(x) = f(x)g(x)
f
f ( x)
, provided g(x) ≠ 0
( x)
g
g ( x)
Quotient:
Example 1 – Defining New Functions Algebraically
Find formulas for the functions f + g, f – g, fg, and f/g
...
f ( x)
x 5; g ( x)
x 3
Solution: We first determine that f has a domain 5, and that g has a domain
These domains overlap, the intersection being the interval 5,
...
5,
Q x
,
M x
f
g
x
f x g x
f x
g x
x 5
x 5
x 3
x 3
DR :
DM : x : x
5,
5, x
3
(A colon was used instead of the “straight
line” after x because I was unable to insert a straight line
...
Because we are finding totally different functions, we give each one its own unique name
...
Since we cannot
actually combine a square root with an absolute value, we leave them the way we did
...
The
composition f of g, denoted f ◦ g, is defined by the rule:
(f ◦ g)(x) = f(g(x))
Read as f of g of x NOT as fog!!
The domain of f ◦ g consists of all x-values in the domain of g that map to g(x)-values in
the domain of f
...
State the domain
...
f ( x)
1
x 2 1; g ( x)
x 1
You start with the outside function and replace the inside function wherever you see an x
...
N x
1
g f x
x
2
1
=
2
1 x
1
2
DN : x : x
2 or
,
2
2, 2
2,
Because the parentheses are not really needed, we combine the negative ones
...
b
...
H x
3
f g x
3
1 x3
1 x3 DH :
,
When you cube a cube root, you get just the part under the radical
...
In example 2, two functions were composed to form a new function
...
This is called decomposing functions
...
In other words, y is the answer we got after we substituted g x into f x
...
1
a
...
y
3
x 5x 3
The easiest way to do this is to let f x be the parent function and let g x be the part you
actually see in the problem (because you substituted it in place of the x in the parent function
...
f x
g x
3x 2
b
...
Feel free to try and come up with
other answers which could work
...
Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second
...
After t seconds, the volume will be 44 t
...
What is the radius of
the balloon after 5 seconds? (You will substitute 44 t in for V and then 5 in place of t )
r
3
r
3
r
3
r
3V
4
3 44t
4
3 44 5
4
3
...
Example 4 – Verifying Pairs in a Relation
Determine which of the ordered pairs (2, -5), (1, 3), and (2, 1) are in the relation defined by
x2y + y2 = 5
...
x2 y
2, 5 : 2
1,3 : 1
2,1 : 2
2
2
y2
5
2
5
3
2
5
3
2
1
1
2
12
5
5
5
Since we did get 5 when we used the ordered pairs 2, 5 and 2,1 , they are in the relation but
because we did not get 5 when we used 1,3 , 1,3 is NOT in the relation
...
** Let us revisit the circle x2 + y2 = 4
...
We will do this by solving the equation for y
...
Title: PRE CALCULUS FUNCTION NOTES
Description: pre cal function notes teaches you how to add multiply divide and subtract functions (FoG)
Description: pre cal function notes teaches you how to add multiply divide and subtract functions (FoG)