function relations and domain of functions
Module 3 : Investigation 2
MAT 170 | Precalculus
September 7, 2016
what is a function ?
Recall from last time, that a function consists of 3 parts :
(1) The set of input values, which we call the domain.
(2) The set of output values, which we call the range.
(3) A rule that assigns each input value in the domain to exactly
one output value.
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how we can express a function
We can express the rule of a function in four different ways :
(1) In words Let f be the function that takes an input of a real number and
outputs the square of the input.
(2) In a table of values x
f(x)
-2
4
-1
1
0
0
4
16
5
25
3
how we can express a function
We can express the rule of a function in four different ways :
(3) By a graph -
(4) By a formula f(x) = x2
10
7.5
5
2.5
-5
-2.5
0
2.5
5
7.5
10
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question 3
Pat was driving his car across Kansas with the cruise control on,
when his gas gauge broke. At the moment the gauge broke, he had 15
gallons of gas in the car’s gas tank and his gas mileage was 42 miles
per gallon (assume he maintains this gas mileage by leaving the
cruise control on).
(a) How many gallons does pat have left after he has driven 84
miles ? 150 miles ?
(b) Define a function f to determine the number of gallons left in
Pat’s tank f(x) in terms of the number of miles driven since the gas
gauge broke x.
(c) What is the domain of f ?
(d) What is the range of f ?
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question 3 - solutions
(a) How many gallons does pat have left after he has driven 84
miles ? 150 miles ?
At 84 miles Pat will have used
84
= 2 gallons.
42
This means he will have 15 − 2 = 13 remaining.
By a similar argument he will have approximately 11.4 gallons
remaining after driving 150 miles.
(b) Define a function f to determine the number of gallons left in
Pat’s tank f(x) in terms of the number of miles driven since the gas
gauge broke x.
f(x) = 15 −
x
42
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question 3 - solutions
(c) What is the domain of f ?
Ask yourself what restrictions there are on the values that can be
used as inputs x for f.
We know that x must be non-negative (Pat cannot drive negative
miles), so x ≥ 0.
Since Pat cannot continue driving after he has run out of gas, we
know that x ≤ a, where a is the number of miles at which Pat
runs out of gas.
To determine a, we solve
15 −
a
a
= 0 −→ 15 =
−→ 630 = a.
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42
Therefore,
Domain : 0 ≤ x ≤ 630
or
[0, 630].
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question 3 - solutions
(d) What is the range of f ?
Ask yourself what are all of the possible outputs f(x) for f.
The tank starts with 15 gallons when x = 0 miles, and ends with
0 gallons when x = 603 miles.
Since every number of gallons between 0 and 15 will occur :
Range : 0 ≤ f(x) ≤ 15
or
[0, 15].
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question 3
Pat was driving his car across Kansas with the cruise control on,
when his gas gauge broke. At the moment the gauge broke, he had 15
gallons of gas in the car’s gas tank and his gas mileage was 42 miles
per gallon (assume he maintains this gas mileage by leaving the
cruise control on).
(e) Let f be as in part (b). What does f(100) represent in this
situation ?
x
represent in this situation ?
42
x
(g) What does 15 −
represent in this situation ?
42
(f) What does
(h) What are the maximum and minimum values that f(x) can
assume in the context of this situation ?
9
question 3 - solutions
(e) Let f be as in part (b). What does f(100) represent in this
situation ?
The number of gallons left in Pat’s tank after driving 100 miles.
(f) What does
x
represent in this situation ?
42
The number of gallons used by Pat to drive x miles.
(g) What does 15 −
x
represent in this situation ?
42
The remaining gallons of gas in Pat’s car after driving x miles.
(h) What are the maximum and minimum values that f(x) can
assume in the context of this situation ?
Maximum : 15
Minimum : 0
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question 3
Pat was driving his car across Kansas with the cruise control on,
when his gas gauge broke. At the moment the gauge broke, he had 15
gallons of gas in the car’s gas tank and his gas mileage was 42 miles
per gallon (assume he maintains this gas mileage by leaving the
cruise control on).
(e1) Let f be as before. Construct a graph of f with the input quantity
(the number of miles driven since the gas gauge broke) on the
horizontal axis, and the output quantity on the vertical axis.
(e2) Explain what the graph of f conveys about the relationship
between the number of miles Pat has driven since his gas gauge
broke x, and the number of gallons of gas left in Pat’s tank f(x).
(j) What does the point (0, 15) on the graph represent in the context
of this situation.
11
question 3 - solutions
50
(e1) Let f be as before. Construct a graph of f with the input quantity
(the number of miles driven since the gas gauge broke) on the
40
horizontal axis, and the output quantity on the vertical axis.
30
20
10
-100
0
100
200
300
400
500
600
700
800
-10
-20
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question 3 - solutions
(e2) Explain what the graph of f conveys about the relationship
between the number of miles Pat has driven since his gas gauge
broke x, and the number of gallons of gas left in Pat’s tank f(x).
The value of f(x) changes at a constant rate with respect to x.
(j) What does the point (0, 15) on the graph represent in the context
of this situation.
When Pat has driven 0 miles (x = 0), the number of gallons of
gas in his tank is 15 (f(0) = 15).
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vertical and horizontal intercepts
When considering the graph of a function, there are two types of
points that tend to be a particular interest.
Definition
The point where the graph of a function f crosses the vertical
axis is called the vertical intercept of f.
The points where the graph of a function f crosses the horizontal axis are called the horizontal intercepts of f.
The vertical intercept (if it exists) is the point (0, b) where f(0) = b.
The horizontal intercepts (if they exist) are points of the form (a, 0)
where f(a) = 0.
Note that there is at most one vertical intercept, while there may be
more than one horizontal intercept. Why ?
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vertical and horizontal intercepts
7.5
y=f (x)
5
2.5
-7.5
-5
-2.5
0
2.5
5
7.5
-2.5
15
vertical and horizontal intercepts
7.5
y=f (x)
5
Vertical intercept
2.5
-7.5
-5
-2.5
0
2.5
5
7.5
-2.5
16
vertical and horizontal intercepts
7.5
y=f (x)
5
Vertical intercept
2.5
-7.5
-5
-2.5
0
2.5
5
7.5
Horizontal intercepts
-2.5
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question 3
Let f be the function giving the number of gallons left in Pat’s tank
f(x), in terms of the number of miles driven since the gas gauge
broke.
(k) Since the vertical intercept of the function f occurs where x = 0,
what general method can you use to determine the vertical intercept
of the graph of a function f.
(l) What is the value of x when f(x) = 0 ? What point of the graph of f
corresponds to f(x) = 0 ? What does this point convey given the
situation ?
(m) Since the horizontal intercept of a function f occurs where
f(x) = 0, what general method can you use to determine the
horizontal intercept of the graph of a function f ?
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question 3 - solutions
(k) Since the vertical intercept of the function f occurs where x = 0,
what general method can you use to determine the vertical intercept
of the graph of a function f.
Input 0 into the function f to determine the output f(0). The vertical intercept will then be the point (0, f(0)).
(l) What is the value of x when f(x) = 0 ? What point of the graph of f
corresponds to f(x) = 0 ? What does this point convey given the
situation ?
We determined that f(x) = 0 when x = 630.
The point on the graph of f corresponding to f(x) = 0 is the horizontal intercept (630, 0).
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question 3 - solutions
(m) Since the horizontal intercept of a function f occurs where
f(x) = 0, what general method can you use to determine the
horizontal intercept of the graph of a function f ?
Solve the equation f(x) = 0 for x. Suppose the solution is x = a.
The horizontal intercept will then be (a, 0).
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