Download 8.4 An Introduction to Functions: Linear Functions, Applications and

8.4
An Introduction to Functions:
Linear Functions, Applications and Models
Suppose we know that gasoline costs $2.15/gallon. Fill in the following:
Number of gallons pumped
Price for this number of gallons
0
1
2
3
4
This table represents a relation. A relation is a set of ordered pairs.
A function is a relation in which for each value of the first component of the ordered pairs
there is exactly one value of the second component.
In a relation, the set of all values of the independent variable (x) is the domain; the set of
all values of the dependent variable (y) is the range.
If the value of the variable y depends on the value of the variable x then y is the dependent
variable and x the independent variable
The graph of a relation is the graph of all its ordered pairs. The graph gives a picture of the
relation which can be used to determine its domain and range. The domain is the horizontal axis
and the range is the vertical axis.
When you have the graph of a relation it is easy to determine if it is a function by the vertical
line test: If a vertical line intersects the graph of a relation in more than one point, then the
relation is NOT a function.
Variations of the definition of a function:
• A function is a relation in which for each value of the first component of the ordered
pairs there is exactly one value of the second component.
• A function is a set of ordered pairs in which no first component is repeated.
• A function is a rule or correspondence that assigns exactly one range value to each
domain value.
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Examples:
For each of the following relations, decide whether it is a function and give the domain
and range.
1. {(1, 1), (1, −1), (2, 4), (2, −4), (3, 9), (3, −9)}
2. {(2, 5), (3, 7), (4, 9), (5, 11)}
3.
4
3
2
1
−4 −3 −2 −1
1
2
3
4
4.
Decide whether the given relation defines y as a function of x. Give the domain.
1. y = x2
2. y = x3
3. x = y 2
√
4. x
2
The following graph shows the relationship between the number of gallons of water in
a small swimming pool and time in hours.
4000
3000
2000
1000
0
0
25
50
75
100
1. What numbers are possible values of the dependent variable?
2. What is the domain and range?
3. For how long is the water level increasing? Decreasing?
4. How many gallons are in the pool after 90 hours?
5. Call this function f . What is f (0)? What does it mean in this example?
Function Notation:
A function is often a rule relating x and y and we often write it as
y = f (x)
and we read it as ”y is a function of ”x”. This is known as function notation. We read
f (x) as ”f of x.”
Example: Let f (x) = 3 + 2x and g(x) = x2 − 2. Find the function values:
1. f (1)
2. f (4)
3. f (−5)
4. g(2)
5. g(0)
6. g(−1)
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Graphing straight lines (linear functions).
A function that can be written in the form
f (x) = mx + b
for real numbers m and b is a linear function. For a linear function written this way m
is the slope and b is the y-intercept. The slope is defined as
m=
rise
run
and b is where the graph crosses the y axis.
Example: Sketch the graph of each linear function. Give the domain and range.
1. f (x) = −2x + 5
2. f (x) = 4x
3. f (x) = 5
An equation that defines y as a function of x is given. Solve for y in terms of x and
replace y with the function notation f (x).
1. y + 2x2 = 3
2. 4x − 3y = 8
Example: Tony Motton runs a copying service in his home. He paid $3500 for the copier
and a lifetime service contract. Each sheet of paper he uses costs $0.01, and he gets paid
$0.05 per copy he makes. Let x represent the number of copies he makes.
1. Express the cost C as a function of x.
2. Express the revenue R as a function of x
3. Determine the value of x for which revenue equals cost
4. Graph y = C(x) and y = R(X) on the same set of axes.
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8.8
Linear Inequalities, Systems, and Linear Programing
Linear Inequalities in Two Variables
A linear inequality in two variables is written as
Ax + By < C
or Ax + By > C,
where A, B and C are real numbers. A linear inequality is essentially a line which divides
the plane into three regions: The two sides of the plane on each side of the line and the
line itself. We are going to be finding solutions which satisfy the inequality so our solution
will be one side of the plane or the other with the line AX + By = C being the boundary.
The steps for graphing a linear inequality are as follows:
Step 1:
Draw the boundary. In this step we graph the line AX + By = C. If
our original problem is stated as < or > then we make the line dashed to indicate that we
don’t want those points.
Step 2: Choose a test point. Pick a point one one side of the line. Make sure you
DO NOT pick a point on the line.
Step 3: Shade the appropriate region. Try your test point in the original equation.
If the equation is true then shade the region that contains the test point. If the equation
is false then shade the other region.
Example 1: Graph −x + 3y ≤ 1
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Example 2: Graph 3x − y > 15
Systems of Inequalities
The steps for graphing a system of inequality are as follows:
Step 1:
Graph each inequality on the same coordinate system. Use the
method described above to do this step.
Step 2:
Find the intersection of the regions of solution. Choose the region
where the two solutions overlap.
Example 3: Graph the solution set of the linear system
x+y ≤ 1
x − y ≥ −5
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Example 4: Graph the solution set of the linear system
2x − y ≥ 1
3x + 2y ≥ 6
Linear Programing
Example 5: Find the maximum and minimum values of of 3x + 5y on the region of
feasible solutions shown.
(5,10)
(2,7)
(6,3)
(1,1)
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Example 6: Elizabeth Owens takes vitamin pills. Each day, she must have at least
16 units of Vitamin A, at least 5 units of Vitamin B, and at least 20 units of Vitamin C.
She can choose between red pills costing $0.10 each that contain 8 units of A, 1 unit of B,
and 2 units of C; and blue pills that cost $0.20 each and contain 2 units of A, 1 of B, and
7 of C. How many of each pill should she take in order to minimize her cost and yet fulfill
her daily requirements?
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8.6
Exponential and Logarithmic Functions, Applications, and Models
Exponential Functions
An exponential function with base b, where b > 0 and b 6= 1, is a function of the form
f (x) = bx ,
where x is any real number.
Use your calculator to find
1. 93/7
2. 142/7
3. (.83)−1.2
8.1
4. 52
√ √
5. ( 7) 3
Graphing Exponential functions:
Example 1: Graph f (x) = 3x , f (x) = 3x and f (x) = ( 13 )x
8
8
6
6
4
4
f (x) = 10x
2
−3
−2
−1
2
f (x) = 3x
1
2
−3
−2
−1
f (x) = ( 13 )x
1
2
Properties of the graph of f (x) = bx
1. The graph wil always contain the point (0,1), since b0 = 1.
2. When b > 1, the graph will rise from left to right. When 0 < b < 1, the graph will
f all from left to right.
3. The x-axis is the horizontal asymptote.
4. The domain is (−∞, ∞) and the range is (0, ∞).
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Compound interest formula: Suppose that a principal of P dollars is invested at an
annual interest rate r (in percent, expressed as a decimal), compounded n times per year.
Then the amount A accumulated after t years is given by the formula
r nt
.
A=P 1+
n
Example: Determine the amount of money that will be accumulated in an account that
pays compound interest, given the initial principal in each of the following:
1. $4292 at 6% compounded annually for 10 years.
2. $4292 at 6% compounded semiannually for 10 years.
3. $4292 at 6% compounded quarterly for 10 years.
4. $4292 at 6% compounded monthly for 10 years.
5. $4292 at 6% compounded daily for 10 years.
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