1. Formalizing Relations and Functions
Section 4-6 Part 2

2. Goals Goal Rubric To find domain and range and use function notation.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
None

4. Domain and Range
When given the domain of a function, you can find the corresponding range of the function by substituting the domain values (x) into the function and solving the range values (y).
Each corresponding domain and range value form an ordered pair (x, y).

5. Example: Find the range of the function for the given domain.
x – 3y = –6; D: {–3, 0, 3, 6}
Step 1 Solve for y since you are given values of the domain, or x.
x – 3y = –6 –x
Subtract x from both sides.
–3y = –x – 6
Since y is multiplied by –3, divide both sides by –3.
Simplify.

6. Example: Continued
Step 2 Substitute the given value of the domain for x and find values of y. x
(x, y) –3
(–3, 1)
(0, 2) 3
(3, 3) 6
(6, 4)

7. Example: Find the range of the function for the given domain.
f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}
Use the given values of the domain to find values of f(x).
f(x) = x2 – 3
(x, f(x)) x –2 –1 1 2
f(x) = (–2)2 – 3 = 1
f(x) = (–1)2 – 3 = –2
f(x) = 02 – 3 = –3
f(x) = 12 – 3 = –2
f(x) = 22 – 3 = 1
(–2, 1)
(–1, –2)
(0, –3)
(1, –2)
(2, 1)

8. Your Turn: Find the range of the function for the given domain.
–2x + y = 3; D: {–5, –3, 1, 4}
Step 1 Solve for y since you are given values of the domain, or x.
–2x + y = 3
+2x x
Add 2x to both sides.
y = 2x + 3

9. Your Turn: Continued –2x + y = 3; D: {–5, –3, 1, 4}
Step 2 Substitute the given values of the domain for x and find values of y. x
y = 2x + 3
(x, y)
y = 2(–5) + 3 = –7 –5
(–5, –7)
y = 2(–3) + 3 = –3 –3
(–3, –3)
y = 2(1) + 3 = 5 1
(1, 5)
y = 2(4) + 3 = 11 4
(4, 11)

10. Your Turn: Find the range of the function for the given domain.
f(x) = x2 + 2; D: {–3, –1, 0, 1, 3}
Use the given values of the domain to find the values of f(x).
f(x) = x2 + 2 x
(x, f(x))
f(x) = (–32) + 2= 11 –3
(–3, 11)
f(x) = (–12 ) + 2= 3 –1
(–1, 3)
f(x) = = 2
(0, 2)
f(x) = =3 1
(1, 3) 3
f(x) = =11
(3, 11)

11. Real-World Range and Domain
When a function describes a real-world situation, every real number is not always reasonable for the domain and range.
For example, a number representing the length of an object cannot be negative, and only whole numbers can represent a number of people.

12. Example: Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each.
Write a function to describe the situation. Find a reasonable domain and range of the function.
Money spent
is $ for each DVD.
f(x) = $ • x
If Joe buys x DVDs, he will spend f(x) = 15x dollars.
Joe only has enough money to purchase 1, 2, or 3 DVDs. A reasonable domain is {1, 2, 3}.

13. Example: Continued 15(1) = 15 15(2) = 30 15(3) = 45
Substitute the domain values into the function rule to find the range values. x 1 2 3
f(x)
15(1) = 15
15(2) = 30
15(3) = 45
A reasonable range for this situation is {$15, $30, $45}.

14. Your Turn:
The settings on a space heater are the whole numbers from 0 to 3. The total of watts used for each setting is 500 times the setting number. Write a function rule to describe the number of watts used for each setting. Find a reasonable domain and range for the function.
Number of
watts used
is times the setting #.
watts
f(x) = • x
For each setting, the number of watts is f(x) = 500x watts.

15. Your Turn: Continued
There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}.
Substitute these values into the function rule to find the range values. x
f(x) 1 2 3
500(0) = 0
500(1) = 500
500(2) =1,000
500(3) =1,500
A reasonable range for this situation is {0, 500, 1,000, 1,500} watts.

16. Your Turn:
Write a function to describe the situation. Find a reasonable domain and range for the function.
A theater can be rented for exactly 2, 3, or 4 hours. The cost is a $100 deposit plus $200 per hour.
f(h) = 200h + 100
Domain: {2, 3, 4}
Range: {$500, $700, $900}

17. Joke Time
Why did the algebra student get so excited after they finished a jigsaw puzzle in only 6 months?
Because on the box it said from 2-4 years.
Why did the algebra student climb the chain-link fence?
To see what was on the other side.
How did the algebra student die drinking milk?
The cow fell on them.

18. Assignment
4-6 Part 2 Exercises Pg : #4 – 18 even