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    Math Aa HL Ch. 3 Notes

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    mohammmed irshad
    A function is a mathematical rule that assigns each element of a domain to a single element of a range. There are two notations for functions: f(x) = ax + b or f: x → ax + b. Composite functions are one function followed by another, such as f(g(x)), while inverse functions find the input when given the output, written as f-1(x). Inverse functions have reflected graphs over the line y = x. Transformations of functions include vertical and horizontal stretches by multiplying x or y values, and vertical and horizontal shifts by adding to x or y values. Rational functions have horizontal or vertical asymptotes where the denominator is zero.

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    Math Aa HL Ch. 3 Notes

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    mohammmed irshad
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    A function is a mathematical rule that assigns each element of a domain to a single element of a range. There are two notations for functions: f(x) = ax + b or f: x → ax + b. Composite functions are one function followed by another, such as f(g(x)), while inverse functions find the input when given the output, written as f-1(x). Inverse functions have reflected graphs over the line y = x. Transformations of functions include vertical and horizontal stretches by multiplying x or y values, and vertical and horizontal shifts by adding to x or y values. Rational functions have horizontal or vertical asymptotes where the denominator is zero.

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    A function is a mathematical rule that assigns each element of a domain to a single element of a range. There are two notations for functions: f(x) = ax + b or f: x → ax + b. Composite functions are one function followed by another, such as f(g(x)), while inverse functions find the input when given the output, written as f-1(x). Inverse functions have reflected graphs over the line y = x. Transformations of functions include vertical and horizontal stretches by multiplying x or y values, and vertical and horizontal shifts by adding to x or y values. Rational functions have horizontal or vertical asymptotes where the denominator is zero.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    38 views2 pages

    Math Aa HL Ch. 3 Notes

    Uploaded by

    mohammmed irshad
    A function is a mathematical rule that assigns each element of a domain to a single element of a range. There are two notations for functions: f(x) = ax + b or f: x → ax + b. Composite functions are one function followed by another, such as f(g(x)), while inverse functions find the input when given the output, written as f-1(x). Inverse functions have reflected graphs over the line y = x. Transformations of functions include vertical and horizontal stretches by multiplying x or y values, and vertical and horizontal shifts by adding to x or y values. Rational functions have horizontal or vertical asymptotes where the denominator is zero.

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    Download as pdf or txt
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    IB Math AA HL Chapter 3 Functions Notes

    A function is a mathematical rule. Although the word “function” is often used for any
    mathematical rule, this is not strictly correct. For a mathematical rule to be a function, each value
    of x can have only one image.

    Vocabulary:
    Domain – the set of numbers that provide the input for the rule.
    Image – the output from the rule of an element in the domain.
    Range – the set of numbers consisting of the images of the domain.
    Co-domain – a set containing the range.
    Function – a rule that links each member of the domain to exactly one member of the range.

    Two types of notations for functions exist, they are f(x) = ax + b or f:x → ax + b

    Review the different types of numbers.

    Composite functions: one function followed by another f  g  x   or f  g  x 


    The order is important f  g  x    g  f  x   accept in rare cases.

    Inverse functions: An inverse function is the reverse of the function. It allows one to find the
    input when they start with the output. f 1  x  means the inverse of f  x 

    How to find an inverse:


    1. Check that an inverse function exists for the given domain.
    2. Rearrange the function so that the subject is x.
    3. Interchange x and y.

    The graph of inverse functions.


    The graph is the reflection over the y = x line.
    If a point on the graph is (x, y) then the point on the inverse is (y, x).
    Reciprocal functions Absolute value function or piecewise function
    1  x, x  0
    f  x  f  x  x or f  x  
    x   x, x  0
    Vertical asymptotes
    x ≠ 0 or denominator ≠ 0

    Drawing a graph:
    Roots – value of x when y = 0
    y-intercept – the value of y when x = 0
    turning points – vertex or where the graph changes direction
    vertical asymptotes – when y is not defined
    horizontal asymptotes – when x → ∞

    Transformations:
    For kf(x), each y-value is multiplied by k and so this creates a vertical stretch.
    For f(kx), each x-value is multiplied by k and so this creates a horizontal stretch.
    For f  x   k , k is added to each y-value and so the graph is shifted vertically.
    For f  x  k  , k is added to each x-value and so the graph is shifted horizontally.
    For  f  x  , each y-value is multiplied by -1 and so each point is reflected in the x-axis.
    For f   x  , each x-value is multiplied by -1 and so each point in reflected in the y-axis.

    g  x a bx  c
    Rational Functions f  x   specifically f  x   and f  x  
    h x  px  q px  q
    a 1
    f  x  this is a transformation of a reciprocal function f  x  
    px  q x

    bx  c a
    f  x  this is similar to f  x   but it has a horizontal asymptotes.
    px  q px  q
    b b
    Horizontal asymptotes y  as when x → ∞, y 
    p p
    Even and Odd functions:
    A function is odd. if for all x in the domain, f(-x) = -f(x). Symmetric about the origin
    A function is even, if for all x in the domain, f(-x) = f(x). Symmetric across the y-axis

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