Precalculus 1.1 Functions and Function Notation

range

the set of second numbers of the ordered pairs in a relation or the output value

Given a table of input and output values, determine whether the table represents a function.

1) Identify the input and output values. 2) Check to see if each input value is paired with only one output value. If so, the table represents a function.

HOW TO Given a relationship between two quantities, determine whether the relationship is a function.

1) Identify the input values and the output values. 2) If each input value leads to only one output value, classify the relationship as a function. 3) If any input value leads to two or more outputs, do not classify the relationship as a function.

A function𝑓

A function is a relation in which each possible input value leads to exactly one output value. It assigns a single value in the range to each value in the domain. We say "the output is a function of the input." In other words, no x-values are repeated.

input value, or independent variable

Each value in the domain, which is often labeled with the lowercase letter 𝑥.

output value, or dependent variable

Each value in the range, which is often labeled lowercase letter 𝑦.

Determining If Class Grade Rules Are Functions In a math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? % grade 0-56 57-61 62-66 67-71 72-77 78-86 87-91 92-100 Grade pt avg. 0.0 1.0. 1.5. 2.0. 2.5. 3.0. 3.5. 4.0

For a % grade earned, there is a grade pt avg, so the grade pt avg is a function of the % grade. In other words, if we input the % grade, the output is a specific grade pt avg. In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

Is {(1,2),(2,4),(3,6),(4,8),(5,10)} a function?

For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, {1, 2, 3, 4, 5}, is paired with exactly one element in the range, {2, 4, 6, 8, 10}.

Does Table 8 represent a function?

Input. Output 1. 10 2. 100 3. 1000

the five greatest baseball players of all time in order of rank. Player. Rank Babe Ruth. 1 Willie Mays. 2 Ty Cobb. 3 Walter Johnson. 4 Hank Aaron. 5

Is the rank a function of the player name? Is the player name a function of the rank?

Evaluating Functions at Specific Values Evaluate𝑓(𝑥)=(𝑥*x)+3𝑥−4 a. 2 b. 𝑎 c. 𝑎+ℎ d. 𝑓(𝑎+ℎ)−𝑓(𝑎)ℎ

Replace the𝑥 x in the function with each specified value. a. Because the input value is a number, 2, we can use simple algebra to simplify.𝑓(2)=(2*2)+3(2)−4=4+6−4=6 b. In this case, the input value is a letter so we cannot simplify the answer any further. 𝑓(𝑎)=𝑎*a+3𝑎−4 c. With an input value of 𝑎+ℎ, 𝑓(𝑎+ℎ)=(𝑎+ℎ)*(a+h)+3(𝑎+ℎ)−4=(𝑎*a)+2𝑎ℎ+(ℎ*h)+3𝑎+3ℎ−4 d. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that𝑓(𝑎+ℎ)=(𝑎*a)+2𝑎ℎ+(ℎ*h)+3𝑎+3ℎ−4 and we know that 𝑓(𝑎)=𝑎*a+3𝑎−4 Now we combine the results and simplify. f(a+h) = (𝑎*a+2𝑎ℎ+ℎ*h+3𝑎+3ℎ−4) -(𝑎*a+3𝑎−4) ------ ---------------------------------- h. h = 2ah + h2 + 3h -------------- h = h( 2a +2 + 3) ------------- h = 2a + h +3

Identifying Tables that Represent Functions

Table 5 is a function Input. Output 2. 1 5. 3 8. 6 Table 6 is a function Input. Output -3. 5 0. 1 4. 5 Table 7 is not a function because the input value of 5 corresponds to two different output values. Input. Output 1. 0 5. 2 5. 4 The function represented by Table 5 can be represented by writing 𝑓(2)=1,𝑓(5)=3,and 𝑓(8)=6 Similarly, the statements 𝑔(−3)=5,𝑔(0)=1,and 𝑔(4)=5 represent the function in Table 6.

Look at a set of ordered pairs. {(1,2),(2,4),(3,6),(4,8),(5,10)} The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first. What is the domain and range?

The domain is {1, 2, 3, 4, 5}. The range is {2,4,6,8,10}.

FUNCTION NOTATION

The notation y=f(x)defines a function named𝑓. This is read as "y is a function of x." The letter x represents the input value, or independent variable. The letter y, or f(x), represents the output value, or dependent variable.

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

The number of days in a month is a function of the name of the month, so if we name the function 𝑓, we write days=𝑓(month) or 𝑑=𝑓(𝑚). The name of the month is the input to a "rule" that associates a specific number (the output) with each input. For example,𝑓(March)=31, because March has 31 days. The notation 𝑑=𝑓(𝑚) reminds us that the number of days, 𝑑 (the output), is dependent on the name of the month, 𝑚 (the input).

domain

The set of the first numbers of the ordered pairs in a relation or the input value

The coffee shop menu consists of items and their prices. Is price a function of the item? Is the item a function of the price? item. price plain donut ...................$1.99 chocolate donut ........$1.49 jelly donut .....................$1.99

Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. Therefore, the item is a not a function of price.

h is f of a

We name the function f, height is a function of age

f(a)

We name the function f; the expression is made as "f of a"

h = f(a)

We use parentheses to determine function input

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value. When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function's formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

A function𝑁=𝑓(𝑦) gives the number of police officers, 𝑁,in a town in year 𝑦. What does 𝑓(2005)=300 represent?

When we read𝑓(2005)=300, we see that the input year is 2005. The value for the output, the number of police officers (𝑁), is 300. Remember, 𝑁=𝑓(𝑦). The statement𝑓(2005)=300 tells us that in the year 2005 there were 300 police officers in the town.

Instead of a notation such as 𝑦=𝑓(𝑥), could we use the same symbol for the output as for the function, such as 𝑦=𝑦(𝑥), meaning "y is a function of x?"

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as𝑓, which is a rule or procedure, and the output 𝑦 we get by applying𝑓 to a particular input 𝑥. This is why we usually use notation such as 𝑦=𝑓(𝑥), 𝑃=𝑊(𝑑), and so on.

relation

a set of ordered pairs

Is {(odd,1),(even,2),(odd,3),(even,4),(odd,5)} a function?

each element in the domain, {even, odd} is not paired with exactly one element in the range, {1, 2, 3, 4, 5}. For example, the term "odd" corresponds to three values from the range, {1, 3, 5} and the term "even" corresponds to two values from the range, {2, 4}. This violates the definition of a function, so this relation is not a function.

Use function notation to express the weight of a pig in pounds as a function of its age in days 𝑑.

y is a function of x so y=f(x) if weight of a pig in lbs is a function of age, in d days then lbs = f ( d )

Given the function ℎ(𝑝)=𝑝*p+2𝑝, solve forℎ(𝑝)=3.

ℎ(𝑝)=3 𝑝*p+2𝑝=3 Substitute the original function h(p)=P*P+2p 𝑝*p+2𝑝−3=0 Subtract 3 from each side. (𝑝+3)(𝑝−1)=0. Factor. If(𝑝+3)(𝑝−1)=0, either (𝑝+3)=0 or (𝑝−1)=0 (or both of them equal 0). We will set each factor equal to 0 and solve for𝑝 (𝑝+3)=0, 𝑝=−3 (p−1)=0, p=1 This gives us two solutions. The output ℎ(𝑝)=3 when the input is either p=1 or p=−3. Figure 6. The graph verifies that ℎ(1)=ℎ(−3)=3 and h(4)=24. Figure 6

Given the functionℎ(𝑝)=𝑝*p+2𝑝, evaluateℎ(4).

ℎ(𝑝)=𝑝*p+2𝑝, ℎ(4)=(4*4)+(2*4) = 25 Therefore, for an input of 4, we have an output of 24.