Pre-Calculus Chapter 1

Functions and Their Graphs

1.2.1 Introduction to Functions

Objectives:

Determine whether relations between

two variables represent a function.

Use function notation and evaluate

functions.

Find the domains of functions.

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Warm Up 1.2.1 Solve Algebraically and Graphically A runner runs at a constant rate of 4.9 miles

per hour.

a. Determine how far the runner can run in 3.1

hours.

b. Determine how long it will take to run a

26.2-mile marathon.

Verbal model: Distance = Rate * Time

Algebraic equation: d = 4.9t3

VocabularyRelationFunctionDomain Range Independent VariableDependent VariableFunction NotationImplied Domain

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Representations of FunctionsIf you pour a cup of coffee, it cools more

rapidly at first, then less rapidly, finally

approaching room temperature.

Since there is one and only one

temperature at any one given time, we

can say that temperature is a

_____________ of time.

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Representation of Functions

Functions can be represented:

Graphically

Algebraically

Numerically

Verbally

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Graphically - Temperature y (°C) as a function of x

(minutes).

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Room Temp.

x (min)

y (deg. C)

Algebraically

Algebraic Equation

y = 20 + 70 (0.8)x

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Numerically

Use equation or

TABLE feature of

graphing

calculator.

x (min.) y (°C)

0 90

5 42.9

10 27.5

15 22.5

20 20.8

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Verbally

If you pour a cup of coffee, it cools more

rapidly at first, then less rapidly, finally

approaching room temperature.

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VariablesIn our coffee example, which is the

dependent variable and which is the

independent variable? Why?

The temperature depends on the amount of

time the coffee has been cooling.

Temperature Dependent

Time Independent

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Domain and RangeDomain

The set of values of the independent variable.

(all “legal” values of x)

Range

The set of values of the dependent variable.

(all “legal” values of y)

What are the domain and range in our

example? 12

Example 1Function or not?

a.

x 2 2 3 4 5

y 11 10 8 5 1

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Example 2Function or not?

a. x2 + y = 1

b. –x + y2 = 1

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Function NotationA function is denoted by the symbol f (x),

“f of x” or “ the value of f at x”.

So, y = f (x).

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Example 3Solve for each if g(x) = –x2 + 4x + 1.

1. g(2) =

2. g(t) =

3. g(x + 2) =

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Domain of a FunctionWe can specify the domain by what it is or by

what it is not.

Explicit Domain

Ex. The set of all real numbers.

Implicit Domain

Ex. x ≠ 0.

This implies all real numbers except x = 0.

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Example 4Let .

What values of x make this function

undefined? Why?

What is the domain of this function?

4

1)(

2 x

xf

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Example 5Let .

What values of x make this function

undefined? Why?

What is the domain of this function?

xxg )(

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Domain in General

The domain of many functions is the

set of all real numbers.

However, we cannot:

Divide by zero

Have a negative number in a square

root (or other even root).

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Domain NotationThe set of all real numbers.

–∞ < x < ∞ or (–∞, ∞)

Exclude a value of x.

x ≠ a or (–∞, a) U (a, ∞)

An interval of x.

a ≤ x < b or [a, b)

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Interval Notation[a, b] a ≤ x ≤ b.

[a, b) a ≤ x < b.

(a, b] a < x ≤ b.

(a, b) a < x < b.

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Example 6

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Homework 1.2.1Worksheet 1.2.1# 1 – 7 odd, 13, 17, 19, 27 – 33 odd

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