Topic 1 Functions

Function

A function is originally the idealization of how a varying quantity depends on another quantity

Interval Notation

A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. Example, [3, 8) is the interval of real numbers between 3 and 8, including 3 and excluding 8.

One to one function

A one-to-one function is a function of which the answers never repeat. Example: The function f(x) = x + 1 is a one-to-one function because it produces a different answer for every input. If the graph crosses the horizontal line more than once it is not an one to one function

Inverse Functions

An inverse function is a function that undoes the action of the another function. A function is the inverse of a function if whenever y = f ( x ) then x = g ( y ) . In other words, applying and then is the same thing as doing nothing. Example If f(x) = 2x - 5, find the inverse. This function passes the Horizontal Line Test which means it is a one to one function that has an inverse. y = 2x - 5 Change f(x) to y. x = 2y - 5 Switch x and y. Solve for y by adding 5 to each side and then dividing each side by 2.

Evaluating functions

Definition To evaluate a function, substitute the input (the given number or expression) for the function's variable (place holder, x). Replace the x with the number or expression. Example Given f(x) = 3x2 + 2x + 1, find f(b). f(b) = 3b2 + 2b + 1

Solving linear systems (three variable) Example First: we add the first and second equation Second: we subtract the third equation from the second Third: we multiply the second equation with 3 on both sides Fourth: Then we add that to the first equation Fifth: we plug this value into the 3x+3y=2 equation in order to determine our y-value Sixth: Last we plug our x- and y-value into any equation in first system in order to determine our z-value

Example: x+2y−z=4 x+2y-z=4 2x+y+z=-2 x-y=-4 2x+y+z=−2 + 2x+y+z=-2 = - x+2y+z=2 = *3 = x+2y+z=2 3x+3y=2 x-y=-4 3x-3y=-12 3x-3y=-12 x= -10/6 3⋅−106+3y=2 y=7/3 + 3x+3y=2 = −5+3y=2 6x = -10 3y=7 x+2y−z=4 z=-1 −10/6+2⋅7/3+z=2 3+z=2

Function composition

Function composition is the point wise application of one function to the result of another to produce a third function Example First we apply f, then apply g to that result f(x) = 2x+3 and g(x) = x^2 (g*f)(x)=(2x+3)^2 If you switch the order you get a different result First we apply g, then apply f to that result (f*g)(x)=2x^2+3

Domain

The domain is the set of all possible x-values which will make the function "work", and will output real y-values. The denominator (bottom) of a fraction cannot be zero The number under a square root sign must be positive in this section Example y= (x+4)square root Domain is x>-4 ​ :

Range

The range is the resulting y-values we get after substituting all the possible x-values. The range of a function is the spread of possible y-values (minimum y-value to maximum y-value) Substitute different x-values into the expression for y to see what is happening. Example y= (x+4)square root Range is y>0 The curve is either on or above the horizontal axis. No matter what value of x we try, we will always get a zero or positive value of y.

Solving linear systems (three variable) Definition

When solving systems of equation with three variables we use the elimination method or the substitution method to make a system of two equations in two variables.