Math Chapter 1.4

checking inverse functions algebraically

evaluate f(f^-1(x)) and f^-1(f(x)): both should be equal to x.

function notation

f(x) = (1 - x) / 2. When using an equation to define a function, you generally isolate the dependent variable (y) on the left, indicating that y is the dependent variable. The independent variable is x, and the name of the function is "f." The symbol f(x) is read as "f or x," and it denotes the value of the dependent variable.

function value

f(x) when x is defined. It lies in the range of f. (ex. the value f(3) is a function value that lies in the range of f. This means that the point (3,f(3)) lies on the graph of f.

functions with and without inverse functions

not every function has an inverse function. in fact, for a function to have an inverse function, it must be one-to-one.

equations with +/- functions?

note that the equations that assign two values (+/-) to the dependent variable for a given value of the independent variable do not define functions of x.

calculate range for y = sqrt(x-1)

observe that sqrt(x-1) is never negative. Moreover, as x takes on the various values in the domain, y takes on all nonnegative values. So that range is the interval y >= 0, or [0,infinity).

Horizontal Line Test

used for one-to-one functions. Geometrically, a function is one-to-one when every horizontal line intersects the graph of the function at most once. So, a graph that represents a one-to-one function must satisfy BOTH the Vertical Line Test and the Horizontal Line Test.

function as a machine

domain x is the input that goes into the function. the output is the range y. (x is the independent variable and y is the dependent variable)

example of finding an inverse function of a more complicated function: f(x) = sqrt(2x-3).

1. write the original function: f(x) = sqrt(2x-3) 2. replace f(x) with y: y = sqrt(2x-3) 3. interchange x and y: x = sqrt(2y-3) 4. square each side: x^2 = 2y - 3 5. add 3 to each side: x^2 + 3 = 2y 6. divide each side by 2: (x^2+3)/2 = y 7. using x as the independent variable, you can write f^-1(x) = (x^2+3)/2 , x >=0 *note that the domain of f^-1 coincides with the range of f.

range and domain of inverse functions

The domain of f must be equal to the range of f^-1, and the range of the f must be equal to the domain of f^-1.

Vertical Line Test

When the graph of a function is sketched, the standard convention is to let the horizontal axis represent the independent variable. When this convention is used, the vertical line test has a nice graphical interpretation. this test states that if every vertical line intersects the graph of an equation at most once, then the equation defines y as a function of x.

illustration of a function

a function can be thought of as a machine that inputs values of the independent variable and outputs values of the dependent variable.

piecewise-defined function

a function defined by two or more equations over a specified domain. (ex. y = { 1- x, x < 1 ; sqrt(x-1), x>=1 }. When x >=1, the function behaves with the domain of the interval [1,infinity). For x <1, the values of 1 - x are positive, so the range of the function is y >=0 or [0,infinity).

one-to-one function

a function is one-to-one when each value of the dependent variable in the range there corresponds exactly one value of the independent variable in the domain. (ex. y = sqrt(x+1).

definition of a function

a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable

how are functions most often specified

although functions can be described by various means such as tables, graphs, and diagrams, they are most often specified by formulas or equations.

calculate domain for y = sqrt(x-1)

because sqrt(x-1) is not defined for x-1 < 0 (that is, for x <1) it follows that the domain of the function is the interval x >= 1 or [1,infinity).

finding an inverse function of a more complicated function

begin by replacing f(x) with y. Then, interchange x and y and solve for y. 1. write the original function 2. replace f(x) with y 3. interchange x and y 4. solve for y

relationship of a function

for a given value of the independent variable, there corresponds exactly one value of the dependent variable

difference quotient

has a special significance in calculus (learn more in chapter 2)

inverse function (informal)

informally, the inverse function of f is another function g that "undoes" what f has done. For instance, subtraction can be used to undo addition, and division can be used to undo multiplication. x (-f->) f(x) (-g->) g(f(x)) = x

how to decide whether an equation defines a function

isolate the dependent variable to the left side, and make sure that for any value of x, there is exactly one (and only one) value of y. So y is a function of x.

an advantage of function notation

it allows you to be less wordy. for instance, instead of asking "what is the value of y when x = 3?" you can ask "what is f(3)?"

definition of composite function

let f and g be functions. the function given by (f o g)(x) = f(g(x)) is the composite of f with g. the domain of f o g is the set of all x in the domain of g such that g(x) is the domain of f (remember that the composite of f with g may not be equal to the composite of g with f)

definition of inverse function

let f and g be two functions such that f(g(x)) = x for each x in the domain of g; and; g(f(x)) = x for each x in the domain of f. Under these conditions, the function g is the inverse function of f. The function g is denoted by f^-1, which is read as "f-inverse." So f(f^-1(x)) = x; and; f^-1(f(x)) = x. The domain of f must be equal to the range of f^-1, and the range of the f must be equal to the domain of f^-1.

implied domain of a function

the domain of a function may be described explicitly, or it may be implied by an equation used to define the function. (ex. in the function y = 1/(x^2-4), it has an implied domain the consists of all real numbers x except x = +/-2. These two values are excluded from the domain because division by zero is undefined. Another type of implied domain is that used to avoid even roots of negative numbers).

domain of a composite function

the domain of f o g is the set of all x in the domain of g such that g(x) is the domain of f

domain

the domain of the function is the set of all values of the independent variable (x) for which the function is defined

ex. inverse function f(x) = 2x

the function f(x) = 2x multiplies each input by 2. to "undo" this function, you need to divide each input by 2. so, the inverse function of f(x) = 2x is f^-1(x) = x/2

graphs of inverse functions

the graphs of f and f^-1 are mirror images of each other.

range

the range of the function is the set of all values taken on by the dependent variable (y).

composite function

the resulting function from a composition.

dependent variable

the value of this variable depends on the value of the other (independent variable)

independent variable

the variable on which the dependent variable depends on

combinations of functions

two functions can be combined in various ways to create new functions. ex. sum: f(x) + g(x) difference: f(x) - g(x) product: f(x)g(x) quotient: f(x)/g(x)

composition

you can combine two functions in yet another way called a composition. the resulting function is called a composite function. f(g(x)). This composition is denoted by f o g and is read as "f composed with g"