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Let's do a quick review of notation concepts from Algebra 1,
and then look at more sophisticated evaluations appropriate for Algebra 2.
Function notation is a manner of "naming" a function. 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.. |
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 Remember: y = f (x).
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).
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 letters used may have significance to a word problem:
If you are comparing the "distance run" in a marathon with the "age of the runner",
you can consider "distance (d) to be a function of age (a)" and name the function d (a).
Set Related Notation: The notation f : X → Ytells 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".)
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Advantages of function notation:
1. |
it allows for individual function names to avoid confusion as to which function is being examined: f (x), g (x), etc.
The graphing calculator does distinctive function naming using subscript numbering such as f1(x) or Y2. |
2. |
it quickly identifies the independent variable in a problem. f (x) = x + 2b + c, where the independent variable is "x". |
3. |
it quickly shows 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." |
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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") |
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| Evaluating Functions: |
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To evaluate a function, simply replace (substitute) the function's variable with the indicated number or expression.
Let's take a look at evaluations that require a bit more sophistication for Algebra 2.
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| 1. |
Given the function f (x) = 5x + 4, find f (2m).
Solution: Substitute 2m into the function in place of x. f (2m) = 5(2m) + 4 = 10m + 4.
Using parentheses will avoid problems.
Notice that the answer is an algebraic expression, not a numeric value.
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2. |
Given f (x) = 3x2 + 2x - 3, find f (2a - 5).
Solution: Parentheses are a MUST is this problem!
Be careful - more algebra work is needed here.
f (2a - 5) = 3(2a - 5)2 + 2(2a - 5) - 3
= 3(4a2 - 20a + 25) + 4a - 10 - 3
= 12a2 - 60a + 75 + 4a - 10 - 3
=
12a2 - 56a + 62
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3. |
Given g ( a) = 9 - a2 and h ( a) = a - 3, express:
| a) g (a) + h (a) |
b) g (a) - h (a) |
| c) g (a) • h (a) |
d)  , g(a) ≠ 0 |
Solution:
a) g (a) + h (a) = (9 - a2) + (a - 3)
= -a2 + a + 6 |
b) g (a) - h (a) = (9 - a2) - (a - 3)
= -a2 - a + 12 |
c) g (a) • h (a) = (9 - a2) • (a - 3)
= -a3 + 3a2 + 9a - 27 |
d) 
a ≠ 3; a ≠ -3 |
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4. |
Given  , express  .
Solution: Warm up your algebraic fraction skills!
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5. |
Given f (x) = x2 - x - 4. Find f (x + h).
Solution: Be careful to replace the x with (x + h). Use parentheses!!!!
(x+h)2 - (x+h) - 4
x2 + 2xh + h2 - x - h - 4
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6. |
Given g(x) = x2 + 1 and h(x) = 5 - x. Express 3•g(5 - x) - 2•h(x2)
Solution: Remember to use parentheses!
3g(5 - x) - 2h(x2) = 3((5 - x)2 + 1) - 2(5 - x2) = 3(x2 - 10x + 25 + 1) - 2(5 - x2)
= 3x2 - 30x + 78 - 10 + 2x2 = 5x2 - 30x + 68 |
7. |
Solution: FYI: This new expression is called the "difference quotient" or average rate of change.
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Things to keep in mind:
Interpreting Negative Answers
When working with word problems, a function may yield a negative answer.
A negative answer often implies a decrease, a loss, or a below-zero state.
Think about what is happening in the word problem you are examining when a result is a negative value. Consider the following example.
EXAMPLE: The weather service posted the following table for Longton City showing the average temperatures for each month in degrees for the second half of the calendar year. Use the function illustrated in the table below, to answer this question.
Find T(10) - T(8).
Solution: From the table, T(10) - T(8) = 52 - 71 = -19.
The negative symbol in this answer indicates that the average temperature from August to October is decreasing. |
Evaluating Functions from Graphs
When a function is represented solely by its graph, you will need to "assume" that certain observable locations are intercepting the grid at integer locations. In the graph below, you can assume (-1,4), (0,1), (1,0) and (-2,-3) are locations on the graph f (x).
(-2,-3) is a bit "questionable" by observation, but it is an actual value.
Example: Given f (x) as shown on the graph,
and g(x) = 4x2 + 6x - 3.
Determine which function has the larger output value when:
a) x = 1
Solution: f (1) = 0 and g(1) = 7 ANS: g(x)
b) x = 0
Solution: f (0) = 1 and g(0) = -3 ANS: f (x)
c) x = -1
Solution: f (-1) = 4 and g(-1) = -5 ANS: f (x)
d) x = -2
Solution: f (-2) = -3 and g(-2) = 1 ANS: g(x)
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For calculator help with
evaluating expressions and functions
click here. |
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