# Functions

 Function 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 x2 shouldn't be a function explain. How do we remedy this? Each root for example √16, has to outputs, 4 and -4The 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 variabledependent 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) 1) Translate c units vertically upward2) Translate c units vertically downward3) Translate c units to the left4) Translate c units to the right5) Reflect in the x-axis6) Reflect in the y-axis Combining functions arithmetically (f+g)(x) = (f-g)(x) = (fg)(x) = (f/g)(x) = (f+g)(x) = f(x) + g(x)(f-g)(x) = f(x) - g(x)(fg)(x) = f(x) * g(x)(f/g)(x) = f(x)/g(x) provided g(x) ≠ 0 Steps to solving f ° g  What is this called Start with an input x and calculate g(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 x0=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 63 is the first iterate of 61.5 is the second iterate of 6 If f(x0) = the first iterate, state the functions for the next three x1 = f(x0) x2 = f(f(x0)) x3 = f(f(f(x0))) x4 = 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 = 2xstep 2: swap all y's with x's so we have x =2ystep 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 ______ symmetryy = xreflectionsaxis 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 inverse function f-1 Authorchikeokjr ID332815 Card SetFunctions Description3.1-3.4 & 3.6 Updated2017-07-17T00:23:50Z Show Answers