Functions

  1. 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
  2. Does every rule or table represent a function?
    Does every graph represent a function?
    • no 
    • no
  3. 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
  4. x2 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
  5. 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
  6. 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,∞)
  7. When asked for the range in for example y = 2x + 6 what do you do?
    solve for x, x = (y-6)/2
  8. T/F: Cube roots are not defined for all real number
    False they are defined for all real numbers
  9. T/F: A vertical represents a function
    F: One x coordinate has multiple y values
  10. 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.
  11. add a card with pictures on pg149
  12. 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)
  13. 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
  14. 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)
  15. 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 upward
    • 2) Translate c units vertically downward
    • 3) Translate c units to the left
    • 4) Translate c units to the right
    • 5) Reflect in the x-axis
    • 6) Reflect in the y-axis
  16. 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
  17. 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
  18. 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
  19. Calculate the rate of change f ° g over time interval t. Which formula do you use?
    Δf ° g/ Δt
  20. 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 6
    • 3 is the first iterate of 6
    • 1.5 is the second iterate of 6
  21. 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))))
  22. 5 rules of multiplying exponents
    Image Upload 1
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. Using graphs, name an easy way to tell which functions are one-to-one
    horizontal line test
  29. 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
  30. Theorem: A function f has an ______ ______ (__) if and only if f is one-to-one
    inverse function f-1
Author
chikeokjr
ID
332815
Card Set
Functions
Description
3.1-3.4 & 3.6
Updated