Name:
Date:
Algebra 2
Function Concept
Perhaps the most important concept in algebra is that of a function, or input-output relationship.
Definition: A function is
A function’s rule can be given in many different ways:
f(x) notation
Often, the symbol for the input is x, and the symbol for the output is f(x), read “f of x.”
With this notation, it’s possible to write an input-output formula as an equation.
Example: f(x) = 2x + 1 means “The rule is: multiply by 2, then add 1.”
Also, you can use the notation to express the output number that goes with an input number.
Example: f(3) = 7 means “When the input is 3, the output is 7.”
Finding the output when given an input
Finding the output for a given input is called evaluating the function. When the function rule is given as an
f(x) equation, the easiest way to evaluate is to substitute the input number in place of x.
Example: For the function f(x) = 2x + 1, evaluate f(3).
(In other words: when the input is 3, what is the output?)
f (x) = 2x + 1
f (3) = 2·3 + 1
f (3) = 7
(substituted 3 in place of x, on both sides)
(did the arithmetic)
Finding the input when given an output
At times we need to do the reverse: we are given an output and must find the corresponding input.
Sometimes this can be accomplished using equation solving.
Example: For the function f(x) = 2x + 1, find an input x that makes f(x) = 11.
(In other words: when the output is 11, what is the input?)
f (x) = 2x + 1
11 = 2x + 1
10 = 2x
5 =x
(substituted 11 as the output, replacing f(x))
(steps to solve the equation for x)
Problems
1.
These questions test your understanding of f(x) notation. Fill in the blanks.
a. The statement f(2) = 9 means that when the input is ______, the output is ______.
b. Suppose that for an input of 4, the output is 17. Then f(_____) = ______.
Name:
Date:
Algebra 2
c. Suppose that (–1, 3) is a point in the graph of f(x). Then f(_____) = ______.
d. If the function rule is f(x) = 4x + 1, then f(–2) = ______.
2.
Answer these questions about the function f(x) = x2 + 1.
a. When the input is –5, what is the output?
b. Evaluate f(6).
c. Find f(–6).
d. Find an input number that would result in an output of 10.
e. Can you find another input number that also results in an output of 10 ?
f. Evaluate f( 23 ).
3.
Answer these questions about the function f(x) = 3  10 .
x
a. Evaluate f(4).
b. Find the input x that makes f(x) = 3000.
c. Evaluate f(–2).
4.
Answer these questions about the function f(x) =
x
 6.
3
a. Write a word description of the function rule. (Tell what’s done to the input to get
the output.)
Name:
Date:
Algebra 2
b. Find f(30). Show an evaluation calculation that leads to your answer.
c. Find the x-value that makes f(x) = 30. Show an equation-solving calculation that
leads to your answer.
d. Find a number you can input to this function, so that the output would be that
same number. Hint: Let n stand for the number you want to find. Put in n for
n
both the input and the output: n = + 6. Now solve for n.
3
5.
Answer these questions about the function f(x) = x · (x – 2).
a. Without using a calculator, find these values: f(5), f(1), and f( 13 ).
b. There are two different inputs x that make the output f(x) equal 0.
Find these two inputs.
6.
Answer these questions about the function f  x  
1
.
x
a. Without using a calculator, evaluate f(–3), f( 14 ), and f( 52 ).
Name:
Date:
Algebra 2
b. Find an input x that makes f(x) = 5.
c. Is there any number that could not be used as an input value for this function?
If so, what?
7.
Here is a word description of a function: “Square root the input number, then add 3.”
a. Write an f(x) formula equation that fits this word description.
b. Find f(16).
c. Can f(–16) be evaluated? If yes, do it. If no, explain why not.
d. What is the smallest possible input value that would make sense for this function?
e. What is the smallest output value that this function could produce?
Functions given by a table or a graph
*When a function is given to you using either a table or a graph, there might not be a
formula that fits the table or graph. That is, the “rule” might simply be that all outputs have
to be looked up using the given table or graph.
Name:
Date:
8.
Algebra 2
Here is an input-output table for a function f(x).
a. What is the value of f(7) ?
b. What x value makes f(x) = 7 ?
c. One of your friends thinks that f(0) = 2.
Another one of your friends thinks that f(0) = 8.
Which friend is right and which friend is wrong? Explain
in a way that the incorrect friend could understand his/her
mistake.
input
x
output
f(x)
0
1
2
3
4
5
6
7
8
5
0
4
7
–1
3
2
d. What x value makes f(x) = 0 ?
9.
A graph representing a function f(x) is shown.
a. What is the value of f(1) ?
b. Find an input x that makes f(x) = 1.
c. Explain why f(4) cannot be evaluated.
d. There are two different inputs that make f(x) = 4. Find both of them.
e. Find an input x that makes f(x) = 0, or
explain why there isn’t such an input.
10. A graph representing a function f(x) is shown.
a. One of the points of this graph is (3, 5).
Fill in the blanks to create an equation that
states a function value:
f( ______ ) = ______.
Name:
Date:
Algebra 2
b. If the input is 6, what is the output?
c. Find f(–3).
d. Find x such that f(x) = 8.
e. There are three different x values that make f(x) = 3. Find all of them.
f. Evaluate f(0).
g. A zero of a function is an input that results in an output of 0.
For example, –2 is one of the zeros of this function, because the graph shows f(–
2) = 0.
What are the other zeros of this function?