Functions Skill 02

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.

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.

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

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.

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

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

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

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

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

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.

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

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

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

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?

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

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

1.2 Functions Summarize Notes Read section 1.2 Homework Pg.24 #7-10, 13, 14, 17-20, 32-35, 42-44, 51, 52, 56-58, 78, 80-83