Let A and B be two nonempty sets. A function from A to B is a rule of correspondence that assigns to each element in set A exactly one element in B
Does every rule or table represent a function?
Does every graph represent a function?
no
no
The set A (x values) in the definition just stated is called the ______ of the function. For these inputs, there outputs called B (y values) and these are referred to as the ______ of the function.
domain
range
x^{2} shouldn't be a function explain. How do we remedy this?
Each root for example √16, has to outputs, 4 and -4
The symbol, √ is defined in algebra as the positive square root only, so √16 = 4
In general, the letter representing elements from the domain (that is, the inputs) is called the _______ _______, for example, x in the equation y = 3x - 2. The letter representing elements from the range (outputs) is called the _______ ________ (y in that equation)
independent variable
dependent variable
If asked to find the domain of the function defined by the equation y = √2x+6, the quantity under the radical sign must be _______. What does this introduce and what is the domain?
non-negative
an inequality, namely 2x+6≥0
[-3,∞)
When asked for the range in for example y = 2x + 6 what do you do?
solve for x, x = (y-6)/2
T/F: Cube roots are not defined for all real number
False they are defined for all real numbers
T/F: A vertical represents a function
F: One x coordinate has multiple y values
Vertical line test
A graph in the x-y plane represents a function of x provided that any vertical line intersects the graph in at most one point.
add a card with pictures on pg149
Conditions for saying function f is increasing/decreasing
Increasing: for all pairs of numbers a and b in the interval, if a < b, then f(a) < f(b)
Decreasing: for all pairs of numbers a and b in the interval, if a < b, then f(a) > f(b)
The average rate of change of a function
The average rate of change of a function f on the interval [a,b] is the slope of the line joining the two points (a,f(a)) and (b,f(b)) aka Δy/Δx
Translation of a graph
we shift in its location such that every point of the graph is moved the same distance in the same direction (size and shape are unchanged)
Property Summary:
1) y = f(x) + c
2) y = f(x) - c
3) y = f(x + c)
4) y = f(x - c)
5) y = -f(x)
6) y = f(-x)
Use g(x) as an input for f; that is calculate f[g(x)]
This is called a composition of functions, in this case, f circle g or f composed with g
What determines the domain of f ° g?
It consists of the inputs that satisfy g(x) for which g(x) is in the domain of f
Calculate the rate of change f ° g over time interval t. Which formula do you use?
Δf ° g/ Δt
In the function f(x) = x/2 when given the input x_{0}=6 the iterates are:
6→3→1.5→.75→.375→.1875→.09375...
The list of numbers are the ______ of ____ under the function f. In the list, the number ____ is the first iterate (of __) and the number ____ is the second iterate (of __) etc
orbit of 6
3 is the first iterate of 6
1.5 is the second iterate of 6
If f(x_{0}) = the first iterate, state the functions for the next three
x_{1} = f(x0)
x_{2} = f(f(x0))
x_{3} = f(f(f(x0)))
x_{4} = f(f(f(f(x0))))
5 rules of multiplying exponents
inverse function [define & state inverse of f(x) = 2x]
swapping the inputs (x values) with the outputs (y values)
inverse of f(x) = f^{-1}(x) = 2/x
Two functions f and g are inverses of one another provided f[g(x)] =
& g[f(x)] =
x for each x in the domain of g
x for each x in the domain of f
How to solve for f(x) = 2x for f^{-1}
step 1: rewrite as y = 2x
step 2: swap all y's with x's so we have x =2y
step 3: solve for y, which should equal 2/x
*Disclaimer: this method does not work for every function with an inverse. We will expand on this as we cover exponential & logarithmic functions in Ch5 and trig and inverse trig functions in Ch6-8
Graphing a function and its inverse always results in a certain type of _______. This will be about the line ___ = ___. The function and its inverse are recognized as _______ & the line is then recognized as the _____ of ______
symmetry
y = x
reflections
axis of symmetry
A function f is one-to-one provided that the following conditions holds for all a and b in the domain of f:
If f (a) = f(b)
then a = b
Using graphs, name an easy way to tell which functions are one-to-one
horizontal line test
A function f is one-to-one if and only if each _______ _____ intersects the graph of y = f(x) in at most one point
horizontal line
Theorem: A function f has an ______ ______ (__) if and only if f is one-to-one