Download Notes: Evaluating Functions

Notes on Evaluating Functions - Page 1
Name_________________________
How to Use a Calculator
1.
Fractions: Fractions are really division problems. Type the numerator, then a division
sign, then the denominator.
Place the fraction inside parentheses.
1
Example: would have to be typed in to look like ( 1  2 ).
2
2.
Exponents: Use the carrot button, or the button that looks like an arrow ( ^ ).
Example: 45 should look like 4 ^ 5 .
3.
Exponents with Negative Bases: When the base is negative, you should place
it in parentheses. For example, -5 raised to the 2nd power should be typed (5)2 .
--------------------------------------------------------------------------------------------------------------------Order of Operations: PEMDAS
 Please Excuse My Dear Aunt Sally
P:
Parentheses
E:
Exponents
M/D: Multiplication/Division (these two operations are of equal importance – they
should be done as they appear from left to right in the expression)
A/S: Addition/Subtraction (these two operations are of equal importance – they should be
done as they appear from left to right in the expression)
If everything is entered correctly, the calculator will follow order of operations.
--------------------------------------------------------------------------------------------------------------------Evaluate without the calculator. Then, use the calculator to check your answer.
1. 47  (3  2) 2
2. 18  45  9
3. 9  3(6  1)2  1
--------------------------------------------------------------------------------------------------------------------Function Notation
Functions are typically named in mathematics with lower-case letters (like f or g).
Suppose a function is named f. Another symbol that means the output of the function is f ( x) .
In fact, the symbols y and f ( x) will be used interchangeably since they both stand for "the
output of function f."
If one desires to know the output of function f when the input of the function is 4, then the idea
could be communicated in several different ways that all mean the same thing:
What is the output of function f if the input is 4?
What is the y-value of function f if the x-value is 4?
What is f (4) ?
So, essentially, the notation f (4) stands for "the output of function f when the input is 4."
Suppose that the table of values to the right is a representation
of function f.
4.
Function f
Input
1
2
3
4
What is f (4) ?
Output
7
12
10
51
5.
What is f (2) ?
--------------------------------------------------------------------------------------------------------------------Let function g be represented by the following list of ordered pairs: {(0,1), (5, 7), (10,12)} .
What is a shorter notation for "the output of function g when the input is 5?"
--------------------------------------------------------------------------------------------------------------------Write out what this notation means: h(2.4) .
--------------------------------------------------------------------------------------------------------------------The process above in which one finds an output when given a specific input is known as
evaluating. You evaluated in function f when you found that the input of 2 produces the output
of 12.
It is easy to evaluate when given the connections of a function in table form, mapping form, or as
a list of ordered pairs. You just take the given input, and write the output connected to that input.
However, the connections between inputs and outputs are more difficult to see in graph form.
--------------------------------------------------------------------------------------------------------------------Consider function j graphed to the right.
6
What is the output when the input is 5?
4
This question could be rewritten as,
"What is the _____-coordinate when the
____-coordinate is 5?
2
-5
So, find the point with an x-coordinate of 5. What is
the y-coordinate of this point?
5
-2
-4
Therefore, j (5)  __________ .
6.
What is j (2) ?
7.
What is j (3) ?
Notes on Evaluating Functions - Page 2
Name_________________________
The connections of a relation are very important. When given a specific input value, the
connections provides the corresponding output value.
In terms of the vending machine example, the connections are like the wiring inside of the
vending machine. The maker of the machine has “wired it” so that when one pushes button ‘4’,
it knows to provide a Pepsi.
Relations and functions sometimes have equations that act as the connections. These are
especially useful when an input can be any real number. Why would it be difficult to show the
connections through the use of a mapping or table if an input can be any real number?
--------------------------------------------------------------------------------------------------------------------Suppose one is given the equation for function k: k ( x)  x  6 .
If you recall, x represents the input, and as was just mentioned, k(x) represents the output.. Thus,
this equation could read, "In function k, the output equals the input plus 6," or, "Add 6 to the
input in order to get the corresponding output."
So, if you wanted to evaluate the function when the input is 4, you could simply substitute 4 into
the equation for x.
k ( x)  x  6
k (4)  4  6  10
Thus, k (4)  10 , or this could be read as "the output is 10 in function k when the input is 4".
--------------------------------------------------------------------------------------------------------------------3
If m( x)  2 x  4 , n( x)  x 2  x , and p ( x) 
, evaluate the following.
x4
8. m(2)
9. n(1)
10. p(8)
--------------------------------------------------------------------------------------------------------------------Homework on Evaluating Functions
Evaluate without the use of a calculator. Then, use a calculator to check your answer.
1.
65 4 2
2.
10  4  8
3.
5  23
4.
64  (9  7)
5.
(3  2)  (4  2)
6.
77  7  7  7
7.
1  (10  5)2
8.
18  3 2
9.
2 2 22 2 2
For Questions 10-12, consider function q graphed on the coordinate plane to the right.
6
10.
What is the range of function q?
11.
Evaluate q(1) .
4
2
-5
12.
5
Is the statement below true or false?
q(2)  q(3)
-2
-4
-6
--------------------------------------------------------------------------------------------------------------------x
If r ( x)  5 x  1 , s( x)  x 2  2 x  3 , and t ( x ) 
, evaluate the following:
x7
13.
r (0)
14.
s (0)
15.
r (10)
16.
s (3)
17.
t (7)
18.
t ( 7)
--------------------------------------------------------------------------------------------------------------------For Questions 19-21, consider the function named u represented in the table below.
19.
What is the domain of function u?
20.
What is u (3) ?
21.
What is u (18) ?
x
0
1
2
3
u(x)
-4
18
-27
15
--------------------------------------------------------------------------------------------------------------------22.
The domain of function w is {2, 4, 6, 8}. Draw a mapping for the function if all of the
following are true:
w(2)  0
w(4)  1
w(6)  1
w(8)  w(6)