
Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a rather lengthy written explanation.
The most popular function notation is f (x) which is read "f of x".
This is NOT the multiplication of f times x.. 


Traditionally, functions are referred to by single letter names, such as f, g, h and so on.
Any letter(s), however, may be used to name a function. Examples:
The f (x) notation is another way of representing the yvalue in a function, y = f (x).
The yaxis may even be labeled as the f (x) axis, when graphing.
Ordered pairs may be written as (x, f (x)), instead of (x, y).
Note: The notation f : X → Y tells us that the function's name is "f " and its ordered pairs are formed by an element x from the set X, and by an element y from the set Y.
(The arrow → is read "is mapped to".)

Advantages of function notation:
1. 
it allows for individual function names to avoid confusion as to which function is being examined.
Names have different letters, such as f (x) and g (x).
The graphing calculator does distinctive function naming with Y1, Y2, ... 
2. 
it quickly identifies the independent variable in a problem. f (x) = x + 2b + c, where the variable is "x". 
3. 
it quickly states which element of the function is to be examined. Find f (2) when f (x) = 3x, is the same as saying, "Find y when x = 2, for y = 3x." 

Equivalent Notations! 
y = 3x + 2 
f (x) = 3x + 2 
f (x) = {(x,y)  y = 3x + 2}
(the vertical bar is read "such that") 
(the bar arrow means the element
"x is mapped/matched to 3x + 2") 





To evaluate a function, substitute the input (the given number or expression) for the function's variable (place holder, x).
Replace the x with the number or expression.

1. 
Given the function f (x) = 3x  5, find f (4).
Solution: Substitute 4 into the function in place of x. f (4) = 3(4)  5 = 7.
This answer can be thought of as the ordered pair (4,7).
The answer may also be referred to as the image of 4 under f (x). 
2. 
Find the value of h (b) = 3b^{2}  2b + 1 when b = 3.
Solution: Substitute 3 into the function in place of b. h (3) = 3(3)^{2}  2(3) + 1 = 34.

3. 
Find g (2w) when g (x) = x^{2}  2x + 1.
Solution: When substituting expressions, like 2w, into a function, using parentheses will help prevent algebraic errors. For this problem, use (2w).
g (2w) = (2w)^{2}  2(2w) + 1 = 4w^{2} 4w +1 (Note: the answer is in terms of w.) 
4. 
Given f (x) = 2x^{2} + 4x  3, find f (2a + 3).
Solution: Be sure to use parentheses!
Be careful  more algebra work is needed here.
f (2a + 3) = 2(2a + 3)^{2} + 4(2a + 3)  3
= 2(4a^{2} + 12a + 9) + 8a + 12  3
= 8a^{2} + 24a + 18 + 8a + 12  3
=
8a^{2} + 32a + 27 
Did you multiply? 


5. 
Given f (x) = x^{2}  x  4. If f (k) = 8, what is the value of k?
Solution: Set the function rule equal to 8 and solve for k.
x^{2}  x  4 = 8
x^{2}  x  12 = 0
(x  4)(x + 3) = 0
x  4 = 0; x + 3 = 0
x = 4; x = 3 
The value of k can be either 4 or 3. 


For calculator help with
evaluating expressions and functions
click here. 

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