1 Functions Full set of Notes No solutions Definition of a Function A rule for which every input number corresponds to only one output number. The graph of a function passes the vertical line test.
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Functions
Full set of Notes
No solutions
Definition of a Function
A rule for which every input number corresponds to only one output number. The graph of a function passes the vertical line test.
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Use your calculator to make a table of values for each function rule, then graph.
a) three times a number b) the square root of a number plus 6 plus 5
c) rule: d) rule:
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Interpreting Graphs
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A speed-time graph.
Find the domain and range.
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A distance-time graph.
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Finding Functional Relationships
1. Find the functional relationship , sketch the graph and state the x intercept, the y intercept, the domain and the range.
2. Find the functional relationship , sketch the graph and state the x intercept, the y intercept, the domain and the range.
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3. Find the functional relationship , sketch the graph and state the x intercept, the y intercept, the domain and the range.
4. Find the functional relationship , sketch the graph and state the x intercept, the y intercept, the domain and the range.
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5. A teacher has to pay $45 shipping and handling charges plus $35 per textbook.
a) Model this problem using a suitable function.
b) Find the cost for 120 books.
c) How many books can be bought for $5000 ?
6. Water at 65 degrees was placed outside on a cold evening. The temperature of the water is dropping at the rate of 2 degrees every 3 minutes.
a) Model this using a suitable function.
b) Find the temperature of the water after 15 minutes of cooling.
c) How long does it take the water to reach the freezing point?
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7. A plant had a height of 7 cm when a special fertilizer was given to it. The plant then began to grow at the rate of 5 cm every 11 days.
a) Model this using a suitable function.
b) How tall will the plant be after 8 weeks?
c) After how many days will the plant be taller than 2 m ?
Applying Linear Functions
These are Examples of Linear Functions
These are NOT Examples of Linear Functions
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1. The amount of water in a tank is given by the linear function
where A is in litres and t in minutes.
a) How much water will there be after 35 minutes?
b) When was there only 400 L? c) When will the tank be empty?
2. Water was flowing into a tank that started with 250 L in it. The water entered at the rate of 17 L every 3 minutes.
a) Model this using a linear function.
b) When will there be 700 L in the tank?
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3. Water was flowing into a tank. After 5 minutes there was 156.8 L in the tank and after 20 minutes there was 241.8 L in the tank.
a) Model this using a linear function.
b) How much water was there in the tank to start with?
c) How much water enters the tank each minute?
Function Notation
1. For f(x) = 3x - 5 evaluate the following
y = 2x + 7 becomes f(x) = 2x + 7 It is to be understood that the "y" is the same as the "f(x)" - y = f(x) , y equals f at x
f(4) =
f(-3) =
f(3/4) =
f(5.21) =
=
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2. For the function evaluate
g(4) =
g(-3) =
g(-2) =
g(3/4) =
g(2.14) =
=
3. For the function f(x) = 2x - 5, determine and simplify the following expressions:
f(m) =
f(2n)=
f(3x)=
f(x+2)=
f(2x+3)=
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4. Suppose h(x) = 5 - 2x, determine and simplify the following expressions:
h(x+1) =
h(2x-3) =
h(2-x) =
h(3x+4)=
5. From the graph determine the following values:
f(12) = f(1) = f(2) =
f(4) = f(6)=
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The Absolute Value Function
The absolute value of a number is the distance that number is away from the origin. Therefore, the absolute value of a number is always positive.
Evaluate the following:
Use your calculator to complete the table of values for each:
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Use your calculator to solve these equations:
1. 2.
3.
Relations
review: function - is a rule that gives a single output number for every input number
definition: relation - is a rule that produces one or more output numbers for every input number
1. the rule: the sum of two numbers is 20
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2. the rule : the square of a number plus a second number is 20
3. the rule : a number plus the square of a second number is 30
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4. the rule : the square of a number plus the square of the second number is 25
5. Determine if each relation is also a function. State the domain and range.
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6. Determine if each relation is also a function. State the domain and range.
7. Determine if each relation is also a function. State the domain and range.
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8. Determine if each relation is also a function. State the domain and range.
9. Determine if each relation is also a function. State the domain and range.
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Find an example of a number that is NOT a valid INPUT for each function.
Evaluate the following when f(x) = 2x + 5 and g(x) = 3x - 2 .
(f + g)(2) =
(f X g)(-3) =
f(g(4)) =
g(f(-2)) =
3f(2) - 4g(5) =
2f(x) + 3g(x) =
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Review: Functions
1. Which of the following is NOT a function?
2. For the functions m(x) = 4x - 3 and n(x) = 7 - 2x
a) evaluate (2m - 5n)(-2)
b) evaluate n(m(4))
c) simplify 5m(x) - 2n(x)
3. Find the domain and range of the graph below.
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4. A large movie theatre had 550 people in it to watch a movie. After the movie ended people left the room at the rate of 50 people every 3 minutes.
a) model the amount of people remaining in the theatre as a function of time (in minutes)
b) how many people are in the room after 15 minutes?
c) after how many minutes will the room be empty?
5. Find the value of the x intercept correctly rounded to the nearest tenth.
-3.2x + 5.6y + 2.6 = 0
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6. Complete the data table for the functions given by
and
State an invalid value for the input for each function.
Answers to the Unit Review Questions
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