PreCalculus Notes: Functions



PreCalculus Notes F-1: Relations and Functions

Relations

A relation is anything that can be represented by a set of ordered pairs. The first elements, usually referred to as x-values for convenience, come from a set called the domain. The second elements, the y-values, form a set called the range.

Ex: {(0, 1), (1, –2), (1, 3), (2, 7)} represents a relation.

Domain:

Range:

Ex: A relation involving polygons: {(Triangle, 3), (Quadrilateral, 4), (Pentagon, 5), …}

Domain:

Range:

Ex: A relation involving jelly beans: {(Red, Cherry), (Orange, Orange), (Yellow, Lemon), (Green, Lime),

(Purple, Grape), (Black, Yucky)}

Domain:

Range:

In PreCalc and beginning Calc, the domain and range normally consist of real numbers. In such cases, relations can often be described by equations.

Ex: Let the relation R be the set of all ordered pairs of real numbers {(x, y) | x = y2}.

a. Determine if each ordered pair is included in R.

1) (2, 4) 2) (9, 3) 3) (9, (3) 4) [pic]

b. Determine if there is a pair in the relation R of the following forms.

1) (x, (0.5) 2) (7, y) 3) ((16, y)

c. What is the domain of R?

d. What is the range of R?

Note: In the example above, the relation is commonly written just x = y2. The {(x, y) | . . . } is understood. (Mathematicians use “understood” to mean “I didn’t bother to write it but you know I meant it.”)

Note: x = y2 could also be written x ( y2 = 0. A relation is often (not always) denoted by F(x, y) = 0.

Relations involving real numbers may be represented in four ways: verbally, graphically, analytically and numerically.

Ex: 1. Verbal: R consists of all points (x, y) that are 5 units from the origin.

2. Graphical: 4. Numerical (a list of ordered pairs or a table):

3. Analytic (i.e., an equation):

Functions

Ex: a. {(Triangle, 3), (Quadrilateral, 4), (Pentagon, 5), …}

If you are told the name of a polygon (given an x-value), can you determine how many sides it has (find

the y-value)?

b. {(Red, Cherry), (Orange, Tangerine), (Yellow, Lemon), (Green, Lime), (Purple, Grape),

(Black, Yucky)}

If you are told the color of a jelly bean (given an x-value), can you determine its flavor (find the

y-value)?

c. {(0, 1), (1, –2), (1, 3), (2, 7)}

If you are given an x-value, can you determine the y-value?

A function is a relation (a set of ordered pairs) where each x-value corresponds to

Ex: A car leaves a 30 mph zone and accelerates at a constant rate of 2 mph/sec until it reaches 60 mph. After exactly five minutes, car and driver are beamed aboard a passing UFO and never heard from again.

a. Make a table of the car's b. Make a graph of the speed as a function of time.

speed v in mph as a

function of time t in

seconds since it began to

accelerate.

|t |v |

|0 | |

|1 | |

|2 | |

|3 | |

|10 | |

|15 | |

|16 | |

|17 | |

| | |

c. Write an equation for v in terms of t.

PreCalculus Notes F-2: Function Notation

Functions are normally denoted by a single letter (like a variable). The letters are typically, but not always, lower case.

Ex: The function f is defined by [pic].

Notes:

1. The name of the function is f.

2. [pic] is pronounced “f of x.” It is a single quantity. It does not mean f times x.

3. [pic] can be thought of as another name for y.

The same function could have been written [pic].

4. The name of the independent variable is not important:

[pic]

[pic]

5. For a given number k, [pic]means the value of f (or y) when x = k.

6. For a number k, [pic] means the x-value(s) that make the value of f be k (i.e., find x so that y = k)

a. Evaluate the following:

1) [pic] = 2) [pic]=

3). [pic] = 4) [pic]=

5) [pic] = 6) [pic] =

7) [pic] =

b. Find the y-intercept of the graph of f.

c. Solve [pic] = 9

d. Find the zeros (aka roots) of f.

Ex: The function g is graphed at right.

a. Evaluate [pic].

b. Evaluate [pic].

c. Solve [pic].

d. State the y-intercept of g.

e. State the zeros (roots) of g.

Ex: The function h is shown in the table at right.

a. Evaluate [pic].

b. Evaluate [pic].

c. Solve [pic].

d. State the y-intercept of h.

e. State the zeros (roots) of h.

PreCalculus Notes F-3: Properties of Functions

Ex: The graph of [pic] is shown at right.

a. Justify that the graph represents a function.

b. What is the domain of the function?

c. What is the range of the function?

d. Evaluate [pic].

e. Solve [pic].

f. Evaluate [pic].

g. Solve [pic].

h. What is the y-intercept of the function?

i. What is/are the root(s) of the function?

j. Where does f have discontinuities?

Note: In PreCalculus terms, a discontinuity is a point where there is some kind of “break” in the graph. We will learn a more precise definition in Calculus.

k. On what interval(s) is f increasing?

Note: in mathematical terms, a function f is increasing on an interval when for any x1 and x2 in the interval, if x2 > x1, then[pic].

l. On what interval(s) is f decreasing?

m. Where does f have a relative maximum (also called a local maximum)?

Note: A function f has a relative (local) maximum at a point c if f(c) ( f(x) for all x in some “neighborhood” of c (the neighborhood is an interval that contains points on both sides of c).

n. Where does f have a local minimum?

o. What is the (absolute or global) maximum value of f?

p. What is the minimum value of f?

Terminology: Extremum (plural, extrema) means either a max or a min.

The value of an extremum means the y-value.

The location of an extremum means the x-value.

Where means give an x-value

What is (the value of) means given a y-value.

PreCalculus Notes F-4: Domain and Range

The natural domain of a function is the set of values for which the function is defined and real.

Defined:

Real:

Ex: For the function[pic], tell if the following numbers are in the natural domain of f and if not, explain why.

a. 0 b. 1 c. 2 d. 3 e. 4

Ex: Find the natural domain of [pic].

Ex: Find the domain of [pic].

Ex: Find the domain of [pic].

Ex: What is the domain of the function [pic]?

In applications and modeling problems, the domain must make sense in the context of the problem.

Ex: A young kid standing on level ground throws a ball. The equation of the ball's path is [pic] where y = height of the ball and x = horizontal distance from the kid. What is the domain of this function?

For PreCalc, the only way we will find the range of a function is graphically.

Ex: Find the range of the function [pic] for x ( [0, 10].

PreCalculus Notes F-6: Compositions

Compositions of functions

The composition of two functions f and g is given by

(1) or (2)

Ex: Let [pic], x ( 2 and [pic], x ( 0.

a. Evaluate [pic] b. Evaluate [pic]

c. Evaluate [pic] d. Evaluate [pic]

d. What is the domain of [pic]? e. What is the domain of [pic]?

The domain of a composition [pic] is

Ex: Let[pic] with a domain of –5 ( x ( 7 and [pic] with a domain of 0 ( x ( 10. Find an expression and a domain for each of the following:

a. [pic] b. [pic].

Recognizing compositions

Ex: For the function [pic], find two functions f and g such that [pic].

Ex: For the function [pic], find two functions f and g such that [pic].

PreCalculus Notes F-7: Transformations I

Reflections

1. Reflection over the y-axis

Ex: Write the equation for the graph of [pic] after a reflection in the y-axis.

Ex: Write the equation for [pic] after a reflection in the y-axis.

2. Reflection over the x-axis.

Ex: Write the equation for the graph of [pic] after a reflection in the x-axis.

Ex: Write the equation for [pic] after a reflection in the x-axis.

3. Reflection over the origin.

Ex: Write the equation for the graph of [pic] after a reflection in the origin.

Ex: Write the equation for [pic] after a reflection in the origin.

Symmetry

1. y-axis symmetry

A graph has y-axis symmetry if the image of the graph after

a reflection in the y-axis is identical to the original graph.

To check for y-axis symmetry, see if the equation is

Ex: Check for y-axis symmetry: [pic]

Ex: Check for y-axis symmetry: [pic]

A function with y-axis symmetry is called an

Note: If a function y = f(x) is an even function (has y-axis symmetry), then

2. x-axis symmetry

A graph has x-axis symmetry if the image of the graph after

a reflection in the x-axis is identical to the original graph.

To check for x-axis symmetry, see if the equation is

Ex: Check for x-axis symmetry: [pic]

3. Origin symmetry

A graph has origin symmetry if the image of the graph after

a reflection in the origin is identical to the original graph.

To check for origin symmetry, see if the equation is

Ex: Check for origin symmetry: [pic]

Ex: Check for origin symmetry: [pic]

A function with origin symmetry is called an

Note: If a function y = f(x) is an odd function (has origin symmetry), then

PreCalculus Notes F-8: Transformations II

Translations

Suppose the graph of [pic] is translated h units to the right

and k units up.

Note: Before solving for y, both directions (horizontal and vertical) seem to work backwards. After solving for y, the horizontal direction still appears to work backwards but the vertical direction seems “normal.”

[pic] moves the graph of f

[pic] moves the graph of f

[pic] or [pic] moves the graph of f

[pic] or [pic] moves the graph of f

Ex: Translate the circle x2 + y2 = 9 four units to the right and five units down.

Ex: Translate the parabola y = x2 – 2x – 1 four units to the right and five units down.

Dilations

Suppose the graph of [pic] is dilated by a factor of a vertically

and a factor of b horizontally.

Again: Before solving for y, both directions (horizontal and vertical) seem to work backwards. After solving for y, the horizontal direction still appears to work backwards but the vertical direction seems “normal.”

[pic] dilates the graph of f

[pic] dilates the graph of f

[pic] or [pic] dilates the graph of f

[pic] or [pic] dilates the graph of f dilates

Ex: Dilate the circle x2 + y2 = 9 by a factor of 5 horizontally and a factor of [pic] vertically.

Ex: Dilate the parabola y = x2 – 2x – 1 by a factor of 2 vertically and a factor of 4 horizontally.

Ex: Write an equation for the graph of the function [pic] after the given transformation.

a. Reflection over the x-axis

b. Translation four units down.

c. Horizontal dilation by a factor of 2.

d. Reflection in the origin.

e. Translation left three and up five.

f. Dilation by a factor of [pic] vertically and [pic] horizontally.

Ex: How does the graph of each of the following compare to the graph of [pic]?

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

g. [pic]

h. y = (x + 4)e(x + 4) – 1

i. [pic]

PreCalculus Practice F-9: Transformations III

Ex: Tell in words how the graphs of each of the following will compare to the graph of y = f(x):

a. y = f(x – 4)

b. y = f(–x)

c. y = 3 f(x)

d. y = f(x) – 4

e. y = f(3x)

f. y = –f(–x)

h. y = f(x + 2) + 3

i. y = [pic]

Ex: The graph of the function [pic] is shown in the first graph below. The other five graphs are transformations of that function. Write their equations.

Ex: The function f, shown in the graph below, consists of two line segments. On the same axes, sketch the graphs of each of the following:

a. y = –f(x)

b. y = f(x – 6) – 4

c. y = f(2x)

d. y = [pic]

e. y = 4f(x)

f. y = [pic]

PreCalculus Notes F-10: Inverses I

Inverse of a relation

Review: A relation is

New: An inverse relation is the relation obtained by

Ex: Let the relation R be {(0, 3), (1, 2), (2, 4), (1, 3), (0, 5)}. Write its inverse.

Note that every relation has an inverse relation.

Inverse of a function

The inverse of a function is defined the same way:

Ex: Find the inverse of the function given in the table:

|x | y |

| 1 |12 |

| 2 | 6 |

| 3 | 4 |

| 4 | 3 |

|x | y |

| 0 | 1 |

| 1 | 4 |

| 2 | 5 |

| 3 | 4 |

| 4 | 1 |

|x | y |

| | |

| | |

| | |

| | |

|x | y |

| | |

| | |

| | |

| | |

| | |

a. b.

Ex: Graph the inverse of the function shown:

a. b.

Ex: Find the inverse of each function:

a. [pic] b. [pic]

Note: Every function has an inverse relation. However, not all of those inverse relations are functions (see all the b examples on previous page). Usually, when we talk about the “inverse of a function” we want the inverse to be another function.

One-to-one functions

Review: A function is a set of ordered pairs (a relation) where

New: A function is one-to-one if

Fact: If a function f is one-to-one then its inverse will be another function, the inverse function f (1. A one-to-one function is said to be invertible.

A function that is not one-to-one is not invertible. (It still has an inverse relation, but it does not have an inverse function).

Inverse functions

Ex: Let f be the function defined by [pic].

a. Is f one-to-one?

b. Find the domain and range of f.

c. Find f (1.

d. Find the domain and range for f (1.

e. Describe in words the “rule” for f and the rule for f (1.

f f (1

1. 1.

2. 2.

3. 3.

f. Find [pic]; call it k. Then find [pic]

g. Evaluate the compositions [pic] and [pic].

Summary of important facts:

1. All one-to-one functions are invertible. In other words, if f is one-to-one, then

2. For one-to-one functions,

In words:

3. To find an inverse function:

a. For a list of ordered pairs or a table:

b. For a graph:

c. For an equation:

4. The range of f (1 is the

AND

The domain of f (1 is the

5. For a one-to-one function and its inverse:

PreCalculus Notes F-11: Inverses II

Ex: A function f with domain [0, 16] is shown in the graph below.

a. What is the range of f?

b. Evaluate f(3).

c. Solve f(x) = 3.

d. Evaluate f (1(3).

e. What is the domain of f -1?

f. Evaluate f (1(10).

g. Evaluate f (1(f(8)).

Ex: A one-to-one function g has domain [(2, 8].

Selected values of g are shown in the table at right.

a. Evaluate [pic].

b. Solve [pic]

c. Evaluate [pic]

d. Evaluate [pic].

Ex: Norman Nerdling’s teacher asked him to find the inverse of the function [pic]. Norman quickly got [pic]. For fun, Norman evaluated [pic] and got 7. This made Norman very happy. For further fun, he evaluated [pic] and got 5. This ruined Norman’s whole day.

a. What did Norman expect to get when he did [pic]?

b. What went wrong?

PreCalculus Review

1. For the function [pic]

a. Solve [pic]algebraically.

b. Find the domain of f.

2. For the function [pic] with domain (–(, (),

a. Write the equation for a new function G(x) that is

1. the translation of g two units to the left and three units down.

2. the dilation of g by a factor of three vertically and a factor of two horizontally.

3. the reflection of g over the y-axis.

b. Without graphing it, determine what kind of symmetry g has, if any, and justify your answer.

c. Find the range of g.

d. Find a domain on which g is one-to-one.

e. Find the inverse of g on the restricted domain form part d.

f. Evaluate [pic]. (If you’re bored, simplify it.)

g. Suppose Doofus brushes his teeth once a day every at 7:00 AM and g(t) represents the number (in millions) of halitosis-causing bacteria in Doofus’s mouth t hours after he brushes his teeth. What is the domain of g? (For simplicity, we will ignore the 15 seconds it takes Doofus to actually brush his teeth.)

PreCalculus Notes: Maxima and Minima

The following definitions are important; they should go into your brains. They are all illustrated in the graph of [pic] with domain [0, d] shown below.

1. A function f is increasing on an interval if [pic] whenever x2 > x1. If a function is going “uphill” from left to right on some interval; in other words, it has positive slope on that interval, then it is increasing.

Note: “The function” refers to the y-values; the y-values are increasing as we move left to right.

Ex: f is increasing on [a, b] and [c, d]; f is decreasing on [0, a] and [b, c].

Note: The endpoints of the intervals are included (if f is continuous there).

2. A function has a relative maximum (or local maximum) at a point [pic]if there is an interval that includes points on both sides of x = xo where for any x ( xo, [pic]. In other words, [pic] is the highest y-value in the “general area” (either side) of xo.

Note: A relative maximum need not be the largest value of ƒ in its entire domain, just the largest value in some "local" part of the graph.

Ex: f has a relative maximum at [pic] since [pic] is greater than [pic] for any other “nearby” values of x on either side of b. f has relative minima at [pic] and [pic].

3. A function has an absolute maximum (or global maximum) of [pic] if [pic] is the largest value of ƒ in the function’s domain. An absolute maximum may occur at a relative maximum or at an endpoint of the domain.

Ex: The absolute maximum value of f is [pic] and occurs at x = d. The absolute minimum value of f is [pic] and occurs at x = c.

Ex: By diligently soliciting door-to-door during the evening of October 31st, Lara was able to accumulate 80 oz. of assorted sweets for later consumption. After much thought, she decided to eat exactly half of her stash each week for as long as it lasted. Write an equation for the amount of candy (in oz.) Lara has left as a function of time in days since Halloween.

Ex: Chuck Rocker sold his collection of moon rocks for $800; this he put into a Certificate of Deposit (CD) which earned 5% annual interest. Find the value of Chuck’s rocks as a function of time.

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y = f(x)

x

y = f(x)

x

1 2 3 4 5 6 7 8 9 10

y = f(x)

y

10

9

8

7

6

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4

3

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1

-1

-2

-3

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x

10

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-3 -2 -1 1 2 3 4 5 6 7 8 9 10

| | |

| (2 | 4 |

| (1 | 1 |

| 0 | (1 |

| 1 | (2 |

| 2 | 0 |

| 3 | 1 |

y = f(x)

y = rx(x)

y = ry(x)

(x, y)

[pic]

h

y

| | |

| (2 | 15 |

| (1 | 12 |

| 0 | 10 |

| 1 | 8 |

| 2 | 7 |

| 3 | 4 |

| 4 | 0 |

| 5 | (2 |

| 7 | (9 |

| 8 | (10 |

f

In

g

Out

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