domain and interval notation
DESCRIPTION
Domain and Interval Notation. Domain. The set of all possible input values (generally x values) We write the domain in interval notation Interval notation has 2 important components: Position Symbols. Interval Notation – Position. Has 2 positions: the lower bound and the upper bound - PowerPoint PPT PresentationTRANSCRIPT
Domain and Interval Notation
Domain The set of all possible input values (generally x
values) We write the domain in interval notation Interval notation has 2 important components:
Position Symbols
Interval Notation – Position Has 2 positions: the lower bound and the
upper bound
[4, 12)Lower Bound• 1st Number• Lowest Possible x-value
Upper Bound• 2nd Number
• Highest Possible x-value
Interval Notation – Symbols
[ ] → brackets
Inclusive (the number is included)
=, ≤, ≥ ● (closed circle)
( ) → parentheses
Exclusive (the number is excluded)
≠, <, > ○ (open circle)
[4, 12) Has 2 types of symbols: brackets and parentheses
Understanding Interval Notation4 ≤ x < 12
Interval Notation:
How We Say It: The domain is 4 to
12 .
On a Number Line:
Example – Domain: –2 < x ≤ 6 Interval Notation:
How We Say It: The domain is –2 to
6 .
On a Number Line:
Example – Domain: –16 < x < –8 Interval Notation:
How We Say It: The domain is –16 to
–8 .
On a Number Line:
Your Turn:
Complete problems 1 – 3 on the “Domain and Interval Notation – Guided Notes” handout
Infinity
Infinity is always exclusive!!! – The symbol for infinity
Infinity, cont.
Negative Infinity Positive Infinity
Example – Domain: x ≥ 4 Interval Notation:
How We Say It: The domain is 4 to
On a Number Line:
Example – Domain: x is Interval Notation:
How We Say It: The domain is to
On a Number Line:
all real numbers
Your Turn:
Complete problems 4 – 6 on the “Domain and Interval Notation – Guided Notes” handout
Restricted Domain When the domain is anything besides (–∞, ∞) Examples:
3 < x 5 ≤ x < 20 –7 ≠ x
Combining Restricted Domains When we have more than one domain restriction,
then we need to figure out the interval notation that satisfies all the restrictions
Examples: x ≥ 4, x ≠ 11 –10 ≤ x < 14, x ≠ 0
Combining Multiple Domain Restrictions, cont.1. Sketch one of the domains on a number line.2. Add a sketch of the other domain.3. Write the combined domain in interval notation.
Include a “U” in between each set of intervals (if you have more than one).
Domain Restrictions: x ≥ 4, x ≠ 11
Interval Notation:
Domain Restrictions: –10 ≤ x < 14, x ≠ 0
Interval Notation:
Domain Restrictions: x ≥ 0, x < 12
Interval Notation:
Domain Restrictions: x ≥ 0, x ≠ 0
Interval Notation:
Challenge – Domain Restriction: x ≠ 2
Interval Notation:
Domain Restriction: –6 ≠ x
Interval Notation:
Domain Restrictions: x ≠ 1, 7
Interval Notation:
Your Turn:
Complete problems 7 – 14 on the “Domain and Interval Notation – Guided Notes” handout
Answers7. 8.
9. 10.
11. 12.
13. 14.
Golf !!!
Answers1. (–2, 7) 6. (–∞,4)2. (–3, 1] 7. (–1, 2) U (2, ∞)3. [–9, –4] 8. [–5, ∞)4. [–7, –1] 9. (–2, ∞)5. (–∞, 6) U (6, 10) U (10, ∞)
Experiment What happens we type the following expressions
into our calculators?
50
05
16
16
*Solving for Restricted Domains Algebraically In order to determine where the domain is
defined algebraically, we actually solve for where the domain is undefined!!!
Every value of x that isn’t undefined must be part of the domain.
*Solving for the Domain Algebraically
In my function, do I have a square root? Then I solve for the domain by: setting the
radicand (the expression under the radical symbol) ≥ 0 and then solve for x
Example Find the domain of f(x).
2x)x(f
*Solving for the Domain Algebraically
In my function, do I have a fraction? Then I solve for the domain by: setting the
denominator ≠ 0 and then solve for what x is not equal to.
Example Solve for the domain of f(x).
1xx6x)x(f
2
*Solving for the Domain Algebraically
In my function, do I have neither? Then I solve for the domain by: I don’t have
to solve anything!!! The domain is (–∞, ∞)!!!
Example Find the domain of f(x).
f(x) = x2 + 4x – 5
*Solving for the Domain Algebraically
In my function, do I have both? Then I solve for the domain by: solving for each
of the domain restrictions independently
Example Find the domain of f(x).
30xxx2)x(f 2
Additional Example Find the domain of f(x).
172x214)x(f
***Additional Example Find the domain of f(x).
6x5x1x510)x(f 2
Additional Example Find the domain of f(x).
41x)x(f
2
Your Turn:
Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout
#8 – Typo!6xx
1)x(f 2
Answers:1. 2.
3. 4.
5.
Answers, cont:6. 7.
8. 9.
10.