Relations and functions Prepared by Doron Shahar Prepared by Doron Shahar Chapter 3 Section 3.1 Relations and functions

Warm-up: page 32 Write the definition of a relation. Prepared by Doron Shahar Prepared by Doron Shahar Warm-up: page 32 Write the definition of a relation. A relation is a correspondence between two sets A and B such that each element of A corresponds to at least one element of B. The domain of a relation is: The range of a relation is: Write the definition of a function. A function is a relation such that each element of the domain corresponds to exactly one element in the range. The set A. The set B.

Relations in diagrams Yes No Yes 1 −2 2 3 4 1 5 −3 −1 2 1 3 Prepared by Doron Shahar Prepared by Doron Shahar Relations in diagrams Domain Range Relation 1 Is it a function? −2 2 Yes No 3 4 1 5 Domain Range −3 Relation Is it a function? −1 2 Yes 1 3

3.1.1(A) Relations in Tables Prepared by Doron Shahar 3.1.1(A) Relations in Tables Determine whether the following represents y as a function of x. x -3 -1 1 3 y 2 Domain Range −3 Is it a function? −1 2 Yes 1 3 If none of the x-values is repeated, then the table represents y as a function of x.

3.1.1(B) Relations in Tables Prepared by Doron Shahar 3.1.1(B) Relations in Tables Determine whether the following represents y as a function of x. x -2 1 y 4 5 2 3 Domain Range Is it a function? 1 −2 2 No 3 4 1 5 If at least one of the x-values is repeated, then the table does not represent y as a function of x.

3.1.2 Relations in Graphs Function Not a function (A) (B) Prepared by Doron Shahar 3.1.2 Relations in Graphs A graph represents the graph of a function if and only if no vertical line passes through the graph more than ________. Determine whether the following represent y as a function of x. Function Not a function (A) (B)

3.1.3 Relations in Equations Prepared by Doron Shahar 3.1.3 Relations in Equations Determine whether the following represent y as a function of x. Steps: 1) Solve for y 2) Plug in trial values for x If for some value of x there is more than one y value, then the equation does not represent y as a function of x. If for all values of x there is only one y value, then the equation represents y as a function of x.

3.1.3(C) Relations in Equations Prepared by Doron Shahar 3.1.3(C) Relations in Equations Determine whether the following represents y as a function of x. 1) Solve for y Starting Equation Solution for y 2) Plug in trial values for x If x=5, If x=9, There is a value of x, namely x=9, for which there are two corresponding y values: +2 and −2. y is NOT a function of x.

3.1.3(A) Relations in Equations Prepared by Doron Shahar 3.1.3(A) Relations in Equations Determine whether the following represents y as a function of x. 1) Solve for y Starting Equation Subtract x2 Take square root Solution for y 2) Plug in trial values for x If x=0, There is a value of x, namely x=0, for which there are two corresponding y values: +3 and −3. y is NOT a function of x.

3.1.3(D): Relations in Equations Prepared by Doron Shahar 3.1.3(D): Relations in Equations Determine whether the following represents y as a function of x. 1) Solve for y Starting Equation Divide by x Take cube root Solution for y 2) Plug in trial values for x If x=−7, For each value of x, there is only one corresponding y value. y is a function of x.

3.1.3(B) Relations in Equations Prepared by Doron Shahar 3.1.3(B) Relations in Equations Determine whether the following represents y as a function of x. 1) Solve for y Starting Equation Factor out y Divide by (x−4) Solution for y 2) Plug in trial values for x If x=−3, For each value of x, there is only one corresponding y value. y is a function of x.

Prepared by Doron Shahar 3.1.6 Review (B) Create a table of values for x and y so that y is NOT a function of x. (C) Create a graph which does NOT represent y as a function of x. (A) Create an equation in x and y so that y is NOT a function of x.

Prepared by Doron Shahar Warm-up Write the following using interval notation. −13 < x ≤ −5 r ≥ 3.6 y < −2 All real numbers All real numbers except x = 3 All real numbers except x = −2 and x = 4

Interval notation Mnemonic : 3.6 ≤ r < ∞ Mnemonic: −∞ < y <−2 Prepared by Doron Shahar Interval notation Write the following using interval notation. −13 < x ≤ −5 r ≥ 3.6 y < −2 Endpoint not included Endpoint included Mnemonic : 3.6 ≤ r < ∞ Mnemonic: −∞ < y <−2

Prepared by Doron Shahar Interval notation Write the following using interval notation. All real numbers All real numbers except x = 3 All real numbers except x = −2 and x = 4

3.2.2 Domain and Range from Graphs Prepared by Doron Shahar 3.2.2 Domain and Range from Graphs Find the domain and range of the graph shown. Range: possible y-values Domain: possible x-values

Page 33: Polynomial functions Prepared by Doron Shahar Page 33: Polynomial functions A polynomial function is a function of the form: f(x)=_________________________, where a0, a1, a2, …, an are real numbers and n is a non-negative integer. Examples of polynomial functions: f(x)=3x5+2x3+7x2+9x+12 g(x)=3x4−2x3+x2−x h(x)=x2+6x+9 q(x)=2x−1 r(x)=17 The domain of every polynomial function is___________.

Page 33: Rational functions Prepared by Doron Shahar Page 33: Rational functions A rational function is a function of the form: f(x)=______, where g and h are polynomial functions and h(x) ≠ 0. Examples of rational functions: 3x5+2x3+7x2+9x+12 17 s(x)= t(x)= 3x4−2x3+x2−x 2x−1 The domain of every rational function is ______________________________________________. 17 Example: The denominator of t(x)= is zero only 2x−1 when x=1/2. The domain of t(x) is

Prepared by Doron Shahar Page 33: Root functions A root function is a function of the form: f(x)= _________, where n is an integer such that n ≥ 2. Examples of root functions: f(x)= 3 x f(x)= 4 x f(x)= x f(x)= 2−5x If n is even, the domain of the root function is _________________________. The domain of f(x)= x is If n is odd, the domain of the root function is _____________________________. The domain of f(x)= 3 x is

3.1.5 Finding the Domain of a Function Prepared by Doron Shahar 3.1.5 Finding the Domain of a Function Determine the domain of the following function. (A) What type of function is g(x)? A polynomial function (−∞,∞) What is the domain of a polynomial function? (−∞,∞) What is the domain of g(x)?

3.1.5 Finding the Domain of a Function Prepared by Doron Shahar 3.1.5 Finding the Domain of a Function Determine the domain of the following function. (B) What type of function is h(x)? A rational function Describe the domain of this rational function. (x≠−3) The set of all x except where x+3=0. What is the domain of h(x) in interval notation?

3.1.5 Finding the Domain of a Function Prepared by Doron Shahar 3.1.5 Finding the Domain of a Function Determine the domain of the following function. (C) What type of function is T(x)? A rational function Describe the domain of this rational function? (x≠−2, x≠5) The set of all x except where x2−3x−10=0. What is the domain of T(x)?

3.1.5 Finding the Domain of a Function Prepared by Doron Shahar 3.1.5 Finding the Domain of a Function Determine the domain of the following function. (D) What type of function is R(x)? A root function Describe the domain of this root function. (x ≤ 2/5) The set of all x where 2−5x ≥ 0. What is the domain of R(x) in interval notation?

Prepared by Doron Shahar 3.1.7 Review Determine a possible equation for a function with the given domain. (A) Domain: the set of all real numbers except x=−2 (C) Domain: the set of all real numbers except x=1 and x=−3 (B) Domain: the set of all real numbers greater than or equal to 4

3.1.4 Evaluating a Function at points Prepared by Doron Shahar 3.1.4 Evaluating a Function at points Use the function f(x)=3x2−5x to evaluate the following. Plug in 4 for x. (A)

3.1.4 Evaluating a Function (B) Prepared by Doron Shahar Prepared by Doron Shahar 3.1.4 Evaluating a Function Use the function f(x)=3x2−5x to evaluate the following. (B)

3.1.4 Evaluating a Function (C) (D) Prepared by Doron Shahar 3.1.4 Evaluating a Function Use the previous slides to evaluate the following. (C) (D)

Prepared by Doron Shahar 3.1.4 Evaluating a Function Use the function f(x)=3x2−5x to evaluate the following.