1. Functions
Skill 02

2. Function…
A function is a relation that assigns each element of set A to exactly one element of set B.
Set A is the domain which is the inputs or commonly called the x values.
Set B is the range which is the outputs or commonly called the y values.

3. Characteristics of a Function…
Each element of A must be matched with and element of B.
Some elements of B may not be matched with any element of A.
Two or more elements of A may be matched with the same B.
An element of A cannot be matched with two elements of B.

4. Variables of a function…
The dependent variable will change based on the independent variable.
𝒚= 𝒙 𝟐 𝒚
Dependent variable?
Independent variable? 𝒙

5. Examples…is the equation a function?
Tell whether y is a function of x in the following equations.
𝒙 𝟐 +𝒚=𝟏
−𝒙+ 𝒚 𝟐 =𝟏
Yes, when solved for y it is a function.
No, when solved for y it is not a function.

6. 𝒚= 𝒙 𝟐 +𝟏 𝒇 𝒙 = 𝒙 𝟐 +𝟏 Function Notation…
In function notation the input is still x, but the output is written as f(x) rather than y.
𝒚= 𝒙 𝟐 +𝟏
𝒇 𝒙 = 𝒙 𝟐 +𝟏

7. Evaluating Functions…
𝒈 𝒙 = −𝒙 𝟐 +𝟒𝒙+𝟏
𝒈 𝟐 =
− 𝟐 𝟐 +𝟒 𝟐 +𝟏
=−𝟒+𝟖+𝟏
𝒈 𝟐 =𝟓
𝒈 𝒕 =
− 𝒕 𝟐 +𝟒 𝒕 +𝟏
𝒈 𝒕 =− 𝒕 𝟐 +𝟒𝒕+𝟏

8. Evaluating Functions…
𝒈 𝒙 = −𝒙 𝟐 +𝟒𝒙+𝟏
𝒈 𝒙+𝟐 =
− 𝒙+𝟐 𝟐 +𝟒 𝒙+𝟐 +𝟏
=− 𝒙 𝟐 +𝟒𝒙+𝟒 +𝟒𝒙+𝟖+𝟏
=− 𝒙 𝟐 −𝟒𝒙−𝟒+𝟒𝒙+𝟖+𝟏
=− 𝒙 𝟐 +𝟓
𝒈 𝒙+𝟐 =− 𝒙 𝟐 +𝟓

9. 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒙<𝟎 𝒙−𝟑, 𝒙≥𝟎 Piece-wise Function…
A function defined by two or more equations over a specified domain.
𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒙<𝟎 𝒙−𝟑, 𝒙≥𝟎

10. Evaluate the Piece-wise Function…
𝒇 𝒙 = 𝒙 𝟐 +𝟏, 𝒙<𝟎 𝒙−𝟏, 𝒙≥𝟎
𝒙=−𝟏
𝒙=𝟎
𝟎=𝟎
−𝟏<𝟎
𝒇 𝟎 =𝟎−𝟏
𝒇 −𝟏 = −𝟏 𝟐 +𝟏
𝒇 𝟎 =−𝟏
𝒇 −𝟏 =𝟐

11. Finding the Domain of a Function…
The domain of a function can either be described explicitly, such as saying that 𝑥≠0, or implied, which is the set of all real numbers for which the expression is defined.

12. Finding the Domain of a Function…
𝒇(𝒙)= 𝟏 𝒙 𝟐 −𝟒
𝒇(𝒙)= 𝒙
Denominator cannot be equal to 0.
There cannot be a negative root.
x ≠±2
x ≥0

13. Finding the Domain of a Function…
𝒇:{ −𝟑,𝟎 , −𝟏,𝟒 , 𝟎,𝟐 , 𝟐,𝟐 , 𝟒, −𝟏 }
𝟎,−𝟑 , 𝟒,−𝟏 , 𝟐,𝟎 𝟐,𝟐 , −𝟏,𝟒

14. Finding the Domain of a Function…
𝒉(𝒙)= 𝟏 𝒙+𝟓
𝒈 𝒙 =𝟑 𝒙 𝟐 +𝟒𝒙+𝟓
𝒉(𝒙)= 𝟗− 𝒙 𝟐

15. Application…
The path of a baseball is given the equation 𝒇 𝒙 =−.𝟎𝟎𝟑𝟐 𝒙 𝟐 +𝒙+𝟑, 𝒇 𝒙 = height (feet), 𝒙= distance (feet). Will the ball clear a 𝟏𝟎’ fence 𝟑𝟎𝟎’ away?

16. Difference Quotient… 𝒇 𝒙+𝒉 −𝒇(𝒙) 𝒉 , 𝒉≠𝟎
𝒇 𝒙+𝒉 −𝒇(𝒙) 𝒉 , 𝒉≠𝟎
*Common definition used in calculus.

17. Evaluating a Difference Quotient…
𝒇 𝒙 = 𝒙 𝟐 −𝟒𝒙+𝟕, 𝒇𝒊𝒏𝒅 𝒇 𝒙+𝒉 −𝒇(𝒙) 𝒉

18. 1.2 Functions Summarize Notes Read section 1.2 Homework
Pg.24 #7-10, 13, 14, 17-20, 32-35, , 51, 52, 56-58, 78, 80-83