 Function Notation and Evaluation MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts Function Notation:

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 y-value in a function, y = f (x).
The y-axis 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")  Evaluating Functions:

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) = 3b2 - 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) = x2 - 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 = 4w2- 4w +1 (Note: the answer is in terms of w.)
4.
Given f (x) = 2x2 + 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(4a2 + 12a + 9) + 8a + 12 - 3
= 8a2 + 24a + 18 + 8a + 12 - 3
= 8a2 + 32a + 27
 Did you multiply? 5.
Given f (x) = x2 - x - 4. If f (k) = 8, what is the value of k?

Solution: Set the function rule equal to 8 and solve for k.
 x2 - x - 4 = 8 x2 - 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. NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".