Topic 4 Functions Graphs’ key features: Domain and Range Intercepts Maximum and Minimum Increasing/ Decreasing Intervals End Behavior

What do you know about …? Domain: [-4,6] Range: [-8,5.5] Y-Intercept: (0,0) X-Intercepts: (-3.5,0),(0,0),(5.5,0) Maximum: y= 5.5 Point of Maximum: (6,5.5) Minimum: y=-8 Point of Minimum: (3,-8) Increasing Intervals: (−𝟒,−𝟐)∪(𝟑,𝟒)∪(𝟓,𝟔) Decreasing Intervals: −𝟐,𝟑 End Behavior: As x approaches to 6, y approaches 5.5 OR 𝒙→𝟔, 𝒕𝒉𝒆𝒏 𝒚→𝟓.𝟓 As x approaches to -4, y approaches -3 OR 𝒙→−𝟒, 𝒕𝒉𝒆𝒏 𝒚→−𝟑

Interval notation  

Functions A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) with each input (x). f(x) y x

Vertical Line Test (pencil test) If any vertical line passes through more than one point of the graph, then that relation is not a function. Are these functions? NOPE! (NOT A FUNCTION) FUNCTION! FUNCTION!

Vertical Line Test FUNCTION! NO! NO WAY! FUNCTION!

Function Notation Input Name of Function Output

A function is a set of ordered pairs of numbers (x, y) such that no x –values are repeated. What are the domain and range of a function? The Domain is the set of all possible x-values in a function. The Range is the set of all possible y-values in a function.

Functions The variable “x” that represents a number in the domain of a function f is called an independent variable. The variable “y” that represents a number in the range of f is called a dependent variable. A function can be specified: algebraically: by means of a formula numerically: by means of a table graphically: by means of a graph verbally: by means of a word problem

Definition Example {(3, 6), (2, 8), (5, 3)} Domain All the x-coordinates in the function's ordered pairs Range All the y-coordinates in the function's ordered pairs { 3, 2, 5} { 6, 8, 3}

Graphs The domain of a function is the set of all the x-coordinates in the functions’ graph Domain 3 ≤ x ≤ 12 Range 6 ≤ y ≤ 12

The graph shows the path of a golf ball What is the range of this function? F 0 < y < 100 G 0 ≤ y ≤ 100 H 0 ≤ x ≤ 5 J 0 < x < 5 Answer is G Is this function continuous or discrete? Would it be reasonable to have a range with negative numbers? What about the Domain?

Sometimes you will be asked to determine a REASONABLE domain or range

The average daily high temperature for the month of May is represented by the function t = 0.2n + 80 Where n is the date of the month. May has 31 days. What is a reasonable estimate of the domain? Answer: 1 ≤ n ≤ 31 What is a reasonable estimate of the range Answer: See next slide

Our function rule is: t = 0.2n + 80 Our domain is 1 ≤ n ≤ 31 Our smallest possible n is 1 Our largest possible n is 31 To find the range, substitute 1 into the equation and solve. Then substitute 31 into the equation and solve.

Our function rule is: t = 0.2n + 80 Substitute a 1 t = 0.2n + 80 t = 0.2(1) + 80 t = 0.2 + 80 t = 80.2 Substitute a 31 t = 0.2n + 80 t = 0.2(31) + 80 t = 6.2 + 80 t = 86.2 Reasonable range is 80.2 ≤ t ≤ 86.2

Finding a Function’s Domain If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.!!!

Algebraically Defined Function Example: is a function. In this case the natural domain of the function is the set In interval notation this is

Example: Finding the Domain of a Function Find the domain of the function Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero. We exclude 7 and – 7 from the domain of g. The domain of g is {x│x ≠ -7 or 7}

Algebraically Defined Function Example: is a function. In this case the natural domain of the function consists of all values of z such that Domain of h(z) is:{z│z ≥ –2/3} In interval notation this is

Algebraically Defined Function Is a function represented by a formula. It has the format y = f (x) = “expression in x” Example: is a function.  

3(1)2+2 1 5 -2 14 3(-2)2+2 Given f(x) = 3x2 + 2, find: 1) f(1) = 5 = 14 3(-2)2+2 -2 14

Algebraically Defined Function Example: is a function. Substitute 5 for x Substitute x+h for x

Numerically Specified Function Example: Suppose that the function f is specified by the following table. x 1 2 3.7 4 f (x) 3.01 -1.03 2.22 0.01 Then, f (0) is the value of the function when x = 0. From the table, we obtain f (0) = 3.01 Look on the table where x = 0 f (2) = 2.22 Look on the table where x = 2 If f (x) = 1, then what is the value of x?

Identifying Intercepts from a Function’s Graph To find the x-intercepts, look for the points at which the graph crosses the x-axis. The y-value is zero (a,0) To find the y-intercept, look for the point at which the graph crosses the y-axis. The x-value is zero (0,b) A function can have more than one x-intercept but at most one y-intercept. Why???

Example: Identifying Intercepts from a Function’s Graph Identify the x- and y-intercepts for the graph of f(x). The x-intercepts are (–3, 0) (–1, 0) and (2, 0) The y-intercept is (0, –6)

Increasing, Decreasing, and Constant Functions 1. A function is increasing on an open interval, I, if f(x1) < f(x2) whenever x1 < x2 for any x1 and x2 in the interval. 2. A function is decreasing on an open interval, I, if f(x1) > f(x2) whenever x1 < x2 for any x1 and x2 in the interval. 3. A function is constant on an open interval, I, if f(x1) = f(x2) for any x1 and x2 in the interval.

Example: Intervals on Which a Function Increases, Decreases, or is Constant State the intervals on which the given function is increasing, decreasing, or constant. increasing on decreasing on increasing on

Example: Use Graphs to Locate Relative Maxima or Minima Identify the relative maxima and minima for the graph of f. f has a relative maximum at x = – 1 f has a relative minimum at x = 1

Now Let’s Practice

What is the domain of this function? What is the range of this function? Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5 Answer: Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5

What is the domain of this function? A -1 ≤ x ≤ 5 B -1 ≤ x ≤ 9 C 2 ≤ x ≤ 5 D 0 ≤ y ≤ 9 Students should immediately realize that “D” cannot be the answer because y is the dependent variable and therefore y represents the RANGE, not the DOMAIN. Answer is A

What is the domain of the function shown on the graph? A -2 < y ≤ 2 B -4 ≤ x ≤ 6 C -4 < y ≤ 2 D -2 < x ≤ 6 Students should immediately realize that A and C cannot be the answer. We are looking for domain. Therefore, our variable will be an X, NOT A Y.

Choosing Realistic Domains and Ranges Consider a function used to model a real life situation Let h(t) model the height of a ball as a function of time What are realistic values for t and for height?

Choosing Realistic Domains and Ranges By itself, out of context, it is just a parabola that has the real numbers as domain and a limited range

Choosing Realistic Domains and Ranges In the context of the height of a thrown object, the domain is limited to 0 ≤ t ≤ 4 and the range is 0 ≤ h ≤ 64

Find Domain and Range Consider the rational function Looking at the formula it is possible to see that since the denominator cannot equal zero, we have a restriction on the domain

Range: -1.19 ≤ y < 0 excluded Find Domain and Range Consider what happens to a function when a denominator gets close to zero when x gets very large Then we have an idea about the range of a function Range: -1.19 ≤ y < 0 excluded