Unit 1: Functions Flashcards
Variables
Relationships among quantities
Function
A function f is a rule that assigns to each value x in a set D a unique value denoted f(x)
Domain
The set D of the function
Range
The set of all values of f(x) produced as x varies over the domain
Independent Variable
The variable associated with the domain
Dependent Variable
Belongs to the range
Graph
The set of all points (x,y) in the xy-plane that satisfy the equation y = f(x)
Argument
The expression on which the function works
Vertical Line Test
Every vertical line intersects the graph at most once, otherwise the graph is not a function
Relation
A set of points on a graph that does not correspond to a function
Composite Functions
Given two functions f and g, the composite function f • g is defined by (f • g) (x) = f(g(x))
Symmetric With Respect To The Y-Axis
If whenever the point (x,y) is on the graph, the point (-x,y) is also on the graph
Symmetric With Respect To The X-Axis
If whenever the point (x,y) is on the graph, the point (x,-y) is also on the graph
Symmetric With Respect To The Origin
If whenever the point (x,y) is on the graph, the point (-x,-y) is also on the graph
Even Function
Has the property that f(-x) = f(x) for all x in the domain
Odd Function
Has the property that f(-x) = -f(x) for all x in the domain
Polynomials
Have the form f(x) = anx^n + an-1x^n-1 + … + ax + a0, where the coefficients a0, a1,… are real numbers which cannot equal 0 and the nonnegative integer n is the degree of the polynomial. An nth degree polynomial can have as many n real zeros or roots-values of x at which f(x) = 0
Rational Functions
Are ratios of the form f(x) = p(x)/q(x), where p and q are polynomials
Algebraic Functions
Constructed using the operations of algebra
Exponential Functions
Have the form f(x) = b^x, where the base b cannot equal 1 is a positive real number
Logarithmic Functions
Have the form f(x) = logb x, where b > 0 and b cannot equal 1
Natural Logarithmic Function
Have the form f(x) = ln x, which also has the base b = e
Natural Exponential Function
Have the form f(x) = e^x, with base b = e, where e approx equal to 2.71828
Trigonometric Function
Are sin x, cos x, tan x, cot x, sec x, and csc x
Inverse Trigonometric Functions
The inverse of trigonometric functions
Transcendental Functions
Consist of trigonometric, exponential, and logarithmic functions
Technology
Graphing calculators and software used to produce graphs of functions
Analytical Methods
Pencil-and-paper methods
Linear Function
A straight-line graph that has the equation y = mx + b, where m and b are constants
Piecewise Functions
Functions that have different definitions on different parts of the domain
Piecewise Linear
If all the pieces of a piecewise function are linear
Power Functions
Have the form f(x) = x^n, where n is a positive integer
Root Functions
Have the form f(x) = x^1/n, where n > 1 is a positive integer
Transformations
Given the real numbers a, b, c, and d and the function f, the graph of y = cf(a(x - b)) + d is obtained from the graph of y = f(x) in the following steps
y = f(x) —- horizontal scaling by a factor of abs value of a —- y = f(ax)
y = f(x) —- horizontal shift by b units —- y = f(a(x-b))
y = f(x) —- vertical scaling by a factor of abs value of c —- y = cf(a(x-b)
y = f(x) —- vertical shift by d units —- y = cf(a(x-b)) +d
Properties of Exponential Functions
- Because b^x is defined for all real numbers, the domain of f is x: -infinity < x < infinity. Because b^x > 0 for all values of x, the range of f is y: 0 < y < infinity
- For all b > 0 b^0 = 1 and thus f(0) = 1
- If b > 1, then f is an increasing function of x.
- If 0 < b < 1, then f is a decreasing function of x
The Natural Exponential Function
f(x) = e^x, which has the base e = 2.718281828459…
Inverse Function
Given a function f, it’s inverse (if it exists) is a function f^-1 such that whenever y = f(x), then f^-1(y) = x
One-To-One
A function f is one-to-one on a domain D if each value of f(x) corresponds to exactly one value of x in D.
Horizontal Line Test
Every horizontal line intersects the graph of a one-to-one function at most once
Existence of Inverse Functions
Let f be a one-to-one function on a domain D with a range R. Then f has a unique inverse f^-1 with domain R and range D such that
f^-1(f(x)) = x and f(f^-1(y)) = y
where x is in D and y is in R
Finding an Inverse Function
Suppose f is one-to-one on an interval I. To find f^-1:
- Solve y = f(x) for x. If necessary, choose the function that corresponds to I.
- Interchange x and y and write y = f^-1(x)
Logarithmic Function Base B
For any base b > 0, with b cannot equal 1, the logarithmic function base b, denoted y = logb x, is the inverse of the exponential function y = b^x
Natural Logarithm Function
The inverse of the natural exponential function with base b = e is the natural logarithm function, denoted y = ln x
Inverse Relations for Exponential and Logarithmic Functions
For any base b > 0, with b cannot equal 1, the following inverse relations hold:
- b^logbx = x (for all x > 0)
- logbb^x = x (for all real values of x)
Properties of Logarithmic Functions
- Because the range of b^x is y: 0 < y < infinity, the domain of logb x is x: 0 < x < infinity.
- The domain of b^x is all real numbers, which implies that the range of logb x is all real numbers
- Because b^0 = 1, it follows that logb 1 = 0
- If b > 1, then logb x is an increasing function
Change-of-Base Rules
Let b be a positive real number with b cannot equal 1. Then
b^x = e^xlnb, for all x and logbx = ln x/ln b, for x > 0
More generally, if c is a positive real number with c cannot equal 1, then
b^x = c^xlogcb, for x and logbx = logcx/logcb, for x > 0
Radians (Rad)
The length of the arc associated with theta, denoted s, divided by the radius of the circle r
Standard Position
An angle is in standard position if its initial side is on the positive x-axis and its terminal side on the line segment OP between the origin and P.
Trigonometric Functions
Let P(x, y) be a point on a circle of radius r associated with the angle theta. Then
sin theta = y/r
csc theta = r/y
cos theta = x/r
sec theta = r/x
tan theta = y/x
cot theta = x/y
Identities
Trigonometric functions that have a variety of properties that are true for all angles in the domain
Period of Trigonometric Functions
- The functions sin theta, cos theta, sec theta, and csc theta have a period of 2pi for all theta in the domain
- The functions tan theta and cot theta have a period of pi for all theta in the domain
Amplitude
A vertical stretch of abs value of A
Period
A period of 2pi/abs value of B
Phase Shift
A horizontal shift of C
Vertical Shift
A vertical shift of D
Inverse Sine/Arcsine
y = sin^-1x is the value of y such that x = sin y, where -pi/2 < or equal to y < or equal to pi/2 with a domain of x: -1 < or equal to x < or equal to 1.
Inverse Cosine/Arccosine
y = cos-1 x is the value of y such that x = cos y, where 0 < or equal to y < or equal to pi with a domain of x: -1 < or equal to x < or equal to 1.
Tan and Cotangent Function
y = tan-1 x is the value of y such that x = tan y, where -pi/2 < y < pi/2 with a domain of x: - infinity < x < infinity
y = cot-1 x is the value of y such that x = cot y, where 0 < y < pi with a domain of x: - infinity < x < infinity.
Secant and Cosecant Function
y = sec -1 x is the value of y such that x = sec y, where 0 < or equal to y < or equal to pi, with y cannot equal pi/2 with a domain of x: abs value of x > or equal to 1
y = csc -1 x is the value of y such that x = csc y, where -pi/2 < or equal to y < or equal to pi/2, where y cannot equal 0 with a domain of x: abs value of x > or equal to 1.
Periodic Functions
Functions whose value repeat every interval of some fixed length
Period
The smallest positive real number that has the property of periodic functions
