Algebra 2 2.1 Introduction to Functions
Relation:
Domain:
Range:
Function:
Vertical Line Test: Used to determine whether the relation has at least one element of the domain paired with more than one element of the range. If the vertical line passes through two or more points on the graph (same x value with 2 different y values), then the relation is not a function
a. b.
Example of functions and how to read them:
a. y = 2x b. f(x) = x + 5 c. g(x) = 2x2
Example of evaluating a function:
1. Given f(x) = x + 7 2. Given g(t) = 7t2 + 1
a. f(5) b. f(-2) a. g(2) b. g(0)
c. f(x + 2) c. For what value of x does g(x) = 64?
3. Given s(t) = -3t + 5 4. Given y = | 3x |
a. s(-1) b. s(t – 2) a. Find y if x = -12
Examples:
Evaluate each function for the given values of x.
5. f(x) = 20x – 4 6. f(x) = 5x2
a) f(-2) a) f(-3)
b) f(8) + f(-2) b) f(5) – f(3)
Examples:
State whether the data in each table represents y as a function of x. Then, state the domain and range.
x / y2 / 2
4 / 3
6 / 4
8 / 5
1. 2.
x / y3 / 4
3 / 5
5 / -4
6 / 3
3. 4.
State the domain, range and whether the relation is a function
5. {(1,2) ; (2,2); (3,1); (4,1)} 6. { (1, 1); ( 1, 3); (2,4)}
D: ______D: ______
R: ______R: ______
Recap!
Identify the domain and range of the given relation. Then, determine if the relation is a function.
1. { (-2, 3), (1, 2), (3, -1), (-4, -3) }
2. { (4, -2), (4, 2), (16, -4), (16, 4) }
Evaluate the function for the given values of x.
3. ; f(8)
4. ; f(-3) + f(6)
5. ; f(15)
6. ; f(-4) – f(2)
Rate of Change:
The table below shows the amount that a company charges for a raft rental for up to 8 hours. An initial deposit of $10 is required, plus $5 per hour after.
Time in hours / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8Rental cost in $
1. Which is the independent variable?
2. Which is the dependent variable?
3. What is the rate of change?
4. How much will you spend after 10 hours?
5. If you have spent $100, how long have you rented the raft?
6. Describe the relationship between time and cost.
7. Write a function for this situation.
8. Graph the situation:
x / f(x)-2
-2
1
1
2
Consider the equation:
1. Use this equation to complete this chart:
2. What is the rate of change?
3. What is f(10) ?
4. For what x-value does f(x) = -14?
5. What is the x-intercept? What is the y-intercept?
6. Graph the line:
x / f(x)-2
-½
0
1
2
Consider the equation:
1. Use this equation to complete this chart:
2. What is the rate of change?
3. What is f(6)?
4. For what x-value does f(x) = -4?
5. What is the x-intercept? What is the y-intercept?
6. Graph the line:
Recap of vocabulary from today:
Independent Variable
Dependent Variable
Rate of Change
Slope
y-intercept
x-intercept
Slope-Intercept Form
Practice: Rate of Change
The table below shows the amount that a company charges for a bike rental for up to 8 hours. An initial fee is included in the amounts shown.
Time in hours / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8Rental cost in $ / 10 / 12 / 14 / 16 / 18 / 20 / 22 / 24
a. What is the rate of change? ______
b. What is the cost for 1 hour? ______
c. How much is the initial fee? ______
d. Which is the independent variable? ______
e. Which is the dependent variable? ______
f. Write a function describing this situation: ______
g. How much is it to rent a bike for 32 hours? (use the equation and show all work!) ______
h. Graph the function: