Module 1

Question 1

A Function:

Answer 1

Is a mapping from a set of inputs to a set of outputs with exactly one output for each input

Question 2

If no domain is stated for a function y = f(x):

Answer 2

the domain is considered to be the set of all real numbers x for which the function is defined.

Question 3

When sketching the graph of a function f

Answer 3

Each vertical line may intersect the graph, at most, once

Question 4

How many zeros and y-intercepts does a function have?

Answer 4

A function may have any number of zeros but it will only have one y-intercept.

Question 5

To define the composition g∘f:

Answer 5

the range of f must be contained in the domain of g.

Question 6

Even Functions

Answer 6

are symmetric about the y-axis

Question 7

Odd functions

Answer 7

Are symmetric about the origin

Question 8

Composition of two functions

Answer 8

(g∘f)(x) = g(f(x))

Question 9

Absolute value function

Answer 9

f(x) = |x| = {x,x≥0

{x,x<0

Question 10

composite function

Answer 10

given two functions f and g, a new function, denoted g∘f, such that (g∘f)(x) = g(f(x)).

Question 11

decreasing on the interval I

Answer 11

a function decreasing on the interval I if, for all x1 ,x2 ∈ I, f(x1) ≥ f(x2) if x1 < x2.

Question 12

dependent variable

Answer 12

the output variable for a function

Question 13

domain

Answer 13

the set of inputs for a function

Question 14

even function

Answer 14

a function is even if f(−x) = f(x) for all x in the domain of f.

Question 15

graph of a function

Answer 15

the set of points (x, y) such that x is in the domain of f and y=f(x)

Question 16

increasing on the interval I

Answer 16

a function increasing on the interval I if for all x1,x2∈I,f(x1)≤f(x2) if x1 < x2

Question 17

independent variable

Answer 17

the input variable for a function

Question 18

odd function

Answer 18

a function is odd if f(−x)=−f(x) for all x in the domain of f

Question 19

range

Answer 19

the set of outputs for a function

Question 20

symmetry about the origin

Answer 20

the graph of a function f is symmetric about the origin if (−x,−y) is on the graph of f whenever (x,y) is on the graph.

Question 21

symmetry about the y-axis

Answer 21

the graph of a function f is symmetric about the y-axis if (−x,y) is on the graph of f whenever (x,y) is on the graph

Question 22

table of values

Answer 22

a table containing a list of inputs and their corresponding outputs

Question 23

vertical line test

Answer 23

given the graph of a function, every vertical line intersects the graph, at most, once

Question 24

zeros of a function

Answer 24

when a real number x is a zero of a function f, f(x) = 0

Question 25

The power function f(x)=x^n is an even function or an odd function if

Answer 25

if n is even and n≠0, and it is an odd function if n is odd

Question 26

What is the Domain of the root function f(x) = x^1/n?

Answer 26

has the domain [0,∞) if n is even and the domain (−∞,∞)

if n is odd.

Question 27

What is the domain of the rational function f(x)=p(x)/q(x), where p(x) and q(x) are polynomial functions?

Answer 27

It is the set of x such that q(x)≠0.

{ x| q(x)≠0}

Question 28

A polynomial function f with degree n≥1

Answer 28

satisfies f(x)→±∞ as x→±∞. The sign of the output as x→∞ depends on the sign of the leading coefficient only and on whether n is even or odd.

Question 29

What are some examples of transformations of functions?

Answer 29

Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the x– and y-axes

Question 30

Point-slope equation of a line

Answer 30

y − y1 = m(x − x1)

Question 31

Slope-intercept form of a line

Answer 31

y = mx+b

Question 32

Standard form of a line

Answer 32

ax + by = c

Question 33

Polynomial function

Answer 33

f(x) = a↓n x^n + a↓n−1 x^n−1 + … + a↓1 x + a↓0

Question 34

algebraic function

Answer 34

a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable
x

Question 35

Transcendental function

Answer 35

Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.

Question 36

cubic function

Answer 36

a polynomial of degree 3; that is, a function of the form f(x) = ax^3+bx^2+cx+d, where a≠0

Question 37

degree

Answer 37

for a polynomial function, the value of the largest exponent of any term

Question 38

linear function

Answer 38

a function that can be written in the form f(x)=mx+b

Question 39

logarithmic function

Answer 39

a function of the form f(x)=log↓b(x) for some base b>0,b≠1 such that y=log↓b(x) if and only if b^y=x

Question 40

mathematical model

Answer 40

A method of simulating real-life situations with mathematical equations

Question 41

piecewise-defined function

Answer 41

a function that is defined differently on different parts of its domain

Question 42

point-slope equation

Answer 42

equation of a linear function indicating its slope and a point on the graph of the function

Question 43

polynomial function

Answer 43

a function of the form f(x)=a↓n x^n + a↓n−1 x^n−1 + ⋯ + a↓1 x + a↓0

Question 44

power function

Answer 44

a function of the form f(x)=x^n for any positive integer n≥1

Question 45

quadratic function

Answer 45

a polynomial of degree 2; that is, a function of the form f(x)=ax^2+bx+c where a≠0

Question 46

rational function

Answer 46

a function of the form f(x)=p(x)/qx), where p(x) and q(x) are polynomials

Question 47

root function

Answer 47

a function of the form f(x)=x^1/n for any integer n≥2

Question 48

slope

Answer 48

the change in y for each unit change in x

Question 49

slope-intercept form

Answer 49

equation of a linear function indicating its slope and y-intercept

Question 50

standard form

Answer 50

equation of a linear function with both variable terms set equal to a constant, ax+by=c.

Question 51

transcendental function

Answer 51

a function that cannot be expressed by a combination of basic arithmetic operations

Question 52

transformation of a function

Answer 52

a shift, scaling, or reflection of a function

Question 53

Generalized sine function equation

Answer 53

f(x) = Asin(B(x−α)) + C

Question 54

periodic function

Answer 54

a function is periodic if it has a repeating pattern as the values of x move from left to right

Question 55

radians

Answer 55

for a circular arc of length s on a circle of radius 1, the radian measure of the associated angle θ is s

Question 56

trigonometric functions

Answer 56

functions of an angle defined as ratios of the lengths of the sides of a right triangle

Question 57

trigonometric identity

Answer 57

an equation involving trigonometric functions that is true for all angles θ for which the functions in the equation are defined

Question 58

Reciprocal identities

Answer 58

tanθ = sinθ cotθ = cosθ cscθ = 1 secθ = 1

cosθ sinθ sinθ cosθ

Question 59

Pythagorean identities

Answer 59

sin^2 θ + cos^2 θ = 1 1 + tan^2 θ = sec^2 θ

1 + cot^2 θ = csc^2 θ

Question 60

Addition and subtraction formulas

Answer 60

sin(α±β) = sinα cosβ ± cosα sinβ
cos(α±β) = cosα cosβ ∓ sinα sinβ

Question 61

Double-angle formulas

Answer 61

sin(2θ) = 2sinθ cosθ
cos(2θ) = 2cos^2 θ − 1 = 1 − 2sin^2 θ = cos^2 θ − sin^2 θ

Question 62

Radian measure is

Answer 62

is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180° has a radian measure of π rad.

Question 63

For acute angles θ,

Answer 63

the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is θ.

Question 64

For a general angle θ

Answer 64

let (x,y) be a point on a circle of radius r corresponding to this angle θ. The trigonometric functions can be written as ratios involving x,y, and r.

Question 65

The trigonometric functions are periodic.

Answer 65

The sine, cosine, secant, and cosecant functions have period 2π. The tangent and cotangent functions have period π.

Question 66

horizontal line test

Answer 66

a function f is one-to-one if and only if every horizontal line intersects the graph of f, at most, once

Question 67

inverse function

Answer 67

for a function f, the inverse function f^−1 satisfies

f^−1 (y) = x if f(x) = y

Question 68

inverse trigonometric functions

Answer 68

the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

Question 69

one-to-one function

Answer 69

a function f is one-to-one if f(x1) ≠ f(x2) if x1 ≠ x2

Question 70

restricted domain

Answer 70

a subset of the domain of a function f

Question 71

When is the exponential function y=b^x increasing/decreasing and what is its domain and range

Answer 71

is increasing if b>1 and decreasing if 0<b></b>

Question 72

The logarithmic function y=log↓b (x) is

Answer 72

is the inverse of y=b^x. Its domain is (0,∞) and its range is (−∞,∞)

Question 73

The natural exponential function is

the natural logarithmic function is

Answer 73

The natural exponential function is y=e^x and the natural logarithmic function is y=lnx=log↓e x

Question 74

Given an exponential function or logarithmic function in base a, we can make a change of base to convert this function to any base

Answer 74

b>0,b≠1. We typically convert to base e

.

Question 75

The hyperbolic functions involve combinations of

Answer 75

f the exponential functions e^x and e^−x. As a result, the inverse hyperbolic functions involve the natural logarithm.

Question 76

base

Answer 76

the number b in the exponential function f(x)=b^x and the logarithmic function f(x)=log↓b x

Question 77

exponent

Answer 77

the value x in the expression b^x

Question 78

hyperbolic functions

Answer 78

the functions denoted sinh,cosh,tanh,csch,sech, and coth, which involve certain combinations of e^x and e^−x

Question 79

inverse hyperbolic functions

Answer 79

the inverses of the hyperbolic functions where cosh and sech are restricted to the domain [0,∞); each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

Question 80

natural exponential function

Answer 80

the function f(x)=e^x

Question 81

number e

Answer 81

as m gets larger, the quantity (1+(1/m))^m gets closer to some real number; we define that real number to be e; the value of e is approximately 2.718282