1.2.1 What is the function?
Function Machines
In the previous lesson, you used equations relating two variables (like y = 3 · 2x) to make graphs
and find information. Today you will look more closely at how equations that relate two
variables help establish a function between the variables. You will also learn a new notation to
help represent these functions.
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1-33. ARE WE RELATED?
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Examine the table of input (x)) and output ((y) values below. Is there a relationship between the
input and output values? If so, write an equation for this relationship.
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1-34. FUNCTION MACHINES
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A function works like a machine, as shown in the diagram below. A function is given a name
that can be a letter, such as f or g.. The notation represents the output when x is processed by the
machine. (Note:f(x) is read, “f of x.”) When x is put into the machine, , the value of a function
for a specific x-value,
value, comes out. In this notation, f(x) replaces y. For example, y = 3x – 1
and f(x) = 3x – 1 are equivalent ways to write the same function.
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Numbers are put into the function machine (in this case, f(x) = 3x – 1) one at a time, and then the
function performs the operation(s) on each input to determine each output. For example,
when x = 3 is put into the function f(x) = 3x – 1, the function triples the input and then subtracts 1
to get the output, which is 8. The notation f(3) = 8 shows that when a 3 is input into the function
named f, the output is 8.
a. Calculate the output for f(
f(x) = 3x – 1 when the input is x = 14;; that is, find f(14).
b. Now find the value of f(–10).
10).
c. If the output of this function is 24, what was the input? That is, if f(x)) = 24, then what
was x? Is there more than one possible input?
1-35. Determine the relationship between x and f(x) in the table below and write the
equation.
1-36. Determine the corresponding outputs or inputs for the following functions. If there
is no possible output for the given input, explain why not.
a.
b.
d.
e.
g.
h.
1-37. Examine the function defined at right. Notice that g(2)
(2) = 5. That is, when the
input x is 2, the output g(2) is 5
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When the input is 1, what is the output of the function? That is, find the value of g(1).
a. Likewise, what are g(–1) and ? g(0)
b. What is the input of this function when the output is –2.5? In other words, find the value
of x when g(x) = –2.5. Is there more than one possible solution?
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Angle Relationships
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If two angles have measures that add up to 90°, they are called complementary angles.
angles
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If two angles have measures that add up to 180°, they are called supplementary angles.
angles
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Vertical angles are the two opposite angles formed by a pair of
intersecting lines, such as angles ∠c and ∠g in the diagram at right.
Vertical angles always have equal measure.
When a line, called a transversal (line l), crosses two parallel lines,
then the corresponding angles are equal. For example, in the
diagram at right, ∠m and ∠d form a pair of equal corresponding angles.
When a transversal (line l)) crosses two parallel lines, then the
alternate interior angles are equal. For example, in the diagram
at right, ∠m and ∠f form a pair of alternate interior angles that are equal.
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1-38. Determine the corresponding outputs for the given inputs of the following functions. If
there is no solution, explain why not. Be careful: In some cases, there may be no solution or
more than one possible solution. Homework Help ✎
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a.
b.
c.
d.
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1-39. If f(x) = 2x, then f(4) = 24= 16. Find the value of:
Homework Help ✎
a. f(1)
b. f(3)
c. f(t)
1-40. A line passes through the points A(
A(–3, –2) and B(2, 1). Does it also pass through
the point C(5, 3)? Justify your conclusion. 1-40 HW eTool(Desmos). Homework Help ✎
1-41. Perform the indicated operation to evaluate each expression. Homework Help ✎
.
a.
b.
c.
1-42. Recall what you learned about angle relationships in previous courses. (See the
Math Notes box in this lesson if you need a reminder.) Determine the measures of the labeled
angles. The double arrowheads indicate that the pair of lines is parallel. Homework Help ✎
a.
b.