Lesson 1.2 Functions Essential Question: What is a function? How do you represent a function? What are the characteristics of a function?

Before we start… Classify as a function or relation and explain why: 3,1 , 2,0 , 1,−1 , 0,−2 , 3,4

What is a relation? A set of ordered pairs Relates an input and an output

What is a function? A function f from a set A to set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs). Each x has only one y

What is the domain? What is the range? The set of all values (inputs) of the independent variable for which the function is defined. If x is in the domain of f, then f is said to be defined at x. If x is not in the domain of f, then f is said to be undefined at x. The set of all values (outputs) assumed by the dependent variable (that is, the set of all function values).

This function can be represented by the ordered pairs {(1, 9), (2, 13), (3, 15), (4, 15), (5, 12), (6, 10)}. In each ordered pair, the first coordinate (x-value) is the input and the second coordinate (y-value) is the output.

How do you represent functions? Verbally (words) The output is 1 less than twice the input. Numerically (x/y table) Graphically (coordinate plane) Algebraically (equation) 𝑦=2𝑥−1 x 1 2 3 4 y 5 7

What are characteristics of a function? Each element of A must be matched with an 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 element of B. An element of A (the domain) cannot be matched with two different elements of B.

How do I identify a function? Check to see if each input is matched with only ONE output.

Is either relation a function?

Let 𝐴= 2, 3, 4, 5 and 𝐵= −3, −2, −1, 0, 1 . Which of the following sets of ordered pairs represent functions from set A to set B. a) 2, −2 , 3, 0 , 4, 1 , 5, −1 b) 4, −3 , 2, 0 , 5, −2 , 3, 1 , 2, −1 c) 2, −1 , 3, −1 , 4, −1 , 5, −1 d) 3, −2 , 5, 0 , 2, −3

Functions are often represented by equations involving two or more variables. 𝑦 = 𝑥2 represents the variable y as a function of x In this equation, x is the independent variable and y is the dependent variable. The domain of the function is the set of all values taken on by the independent variable x, and the range of the function is the set of all values taken on by the dependent variable y.

How do you determine if an equation is a function algebraically? Try to solve the equation for y in terms of x.

Determine whether the equation represents y as a function of x. 𝑥 2 +𝑦=1

Determine whether the equation represents y as a function of x. −𝑥+ 𝑦 2 =1

Determine whether the equation represents y as a function of x. 𝑥 2 + 𝑦 2 =8

What is function notation? You write a function by giving it the name f instead of using y “the value of f at x” or “f of x”

Function Values with 𝑓 𝑥 𝑓 0 means evaluate the function when 𝑥=0 𝑓 −3 means evaluate the function when 𝑥=−3 𝑓 7 means evaluate the function when 𝑥=7

Function Notation 𝑦=𝑓 𝑥 f is the name of the function. y is the dependent variable, or output value. x is the independent variable, or input value. 𝑓 𝑥 is the value of the function at x.

How do you evaluate a function? Replace the x with the given number and simplify f (x) is just another name for y. It does not mean f times x.

Let 𝑔 𝑥 =− 𝑥 2 +4𝑥+1 and find the following. b) 𝑔 𝑡 c) 𝑔 𝑥+2

Let 𝑓 𝑥 =10−3 𝑥 2 and find the following. b) 𝑓 −4 c) 𝑓 𝑥−1

What is a piecewise-defined function? A function defined by two or more equations over a specified domain 𝑓 𝑥 = 𝑥−4, 𝑥<−3 2 5 𝑥, 𝑥≥−3

Library of Parent Functions: Absolute Value Function Graph of 𝑓 𝑥 = 𝑥 = 𝑥, 𝑥≥0 −𝑥, 𝑥<0 Domain: −∞, ∞ Range: 0 , ∞ Intercept: 0, 0 Decreasing on −∞, 0 Increasing on 0, ∞

Evaluate the function 𝑓 𝑥 = 3𝑥−4, 𝑥<0 3𝑥+1, 𝑥≥0 a) 𝑓 −2 b) 𝑓 0 c) 𝑓 2

Evaluate the function 𝑓 𝑥 = 𝑥 2 +1, 𝑥<0 𝑥−1, 𝑥≥0 a) 𝑓 −1 b) 𝑓 0 c) 𝑓 7

What is the implied domain? If f is defined by an algebraic expression and the domain is not specified, then the implied domain consists of all real numbers for which the expression is defined.

Library of Parent Functions: Square Root Function Graph of 𝑓 𝑥 = 𝑥 Domain: 0 , ∞ Range: 0 , ∞ Intercept: 0, 0 Increasing on 0, ∞

How do you find the domain of a function? When finding domain on the real number line, there are two things you have to check for: 1. Division by zero 2. Even roots of negative numbers

Find the domain of each function. 𝑓 : {(–3, 0), (–1, 4), (0, 2), (2, 2), (4, –1)}

Find the domain of each function. 𝑔 (𝑥) = –3𝑥2 + 4𝑥 + 5

Find the domain of each function. ℎ 𝑥 = 1 𝑥+5

Find the domain of 𝑔 𝑥 = 4 9− 𝑥 2 .

Find the domain of 𝑔 𝑥 = 𝑥−16 .

Use a graphing utility to find the domain and range of the function 𝑓 𝑥 = 16− 𝑥 2 .

The number N (in millions) of employees in the construction industry in the United States increased in a linear pattern from 2003 through 2006. In 2007, the number dropped, then decreased through 2008 in a different linear pattern. These two patterns can be approximated by the function N 𝑡 = 0.32t + 5.7 , 3  t  6 –0.42t + 10.5, 7  t  8 where t represents the year, with t = 3 corresponding to 2003. Use this function to approximate the number of employees for each year from 2003 to 2008. (Source: U.S. Bureau of Labor Statistics)

The total value V (in billions of dollars) of farm real estate in the United States from 1997 through 2007 can be approximated by the model 𝑉 𝑡 = 50.10𝑡+498.8, 7≤𝑡≤13 232.31𝑡−2013.5, 14≤𝑡≤17 where t = 7 represents 1997. Use this model to approximate the total value of farm real estate in 1999, 2004, and 2006.

A baseball is hit at a point 3 feet above ground at a velocity of 100 feet per second and an angle of 45o. The path of the baseball is given by 𝑦= −0.0032𝑥 2 +𝑥+3, where x and y are measured in feet. Will the baseball clear a 20-foot fence located 280 feet from home plate?

What is the Difference Quotient? One of the basic definitions in calculus employs the ratio 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ , ℎ≠0

For 𝑓 𝑥 = 𝑥 2 −4𝑥+7, find 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ .

For 𝑓 𝑥 = 𝑥 2 −2𝑥+9, find 𝑓 4+ℎ −𝑓 4 ℎ .

What is a function. How do you represent a function What is a function? How do you represent a function? What are the characteristics of a function?

Ticket Out the Door Given 𝑓 𝑥 =2+3𝑥− 𝑥 2 , find 𝑓 𝑥+ℎ −𝑓(𝑥) find the domain