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AS Level
Mathematics 9709 P1

2. Functions

Written by: Tharun Athreya
Formatted by: Dhyaneshwaran V

Index

2.1 Definition of a function
2.2 Composite functions
2.3 Inverse functions
2.4 The graph of a function and its inverse
2.5 Transformations of functions
2.6 Combined transformations

2.1 Definition of a function

  • A function is a relation that uniquely associates members of one set with the members of another set.
    • Functions can also be called mappings.
  • Mappings can be:
    • ONE-TO-ONE
    • MANY-TO-ONE
    • ONE-TO-MANY
    • MANY-TO-MANY
  • Functions can be either a one-one function or a many-one function.
  • A one-one function has one output value for each input value.
  • A many-one function has one output value for each input value, but each output value can have more than one input value.
  • The set of input values for a function is called the domain of the function.
  • The set of output values for a function is called the range of the function.

2.2 Composite functions

  • When one function is followed by another function, the resulting function is called a composite function.
  • \( fg(x) \) means the function \( g \) acts on \( x \) first, then \( f \) acts on the result.
  • There are three important points to remember about composite functions:
    • \( fg \) only exists if the range of \( g \) is contained within the domain of \( f \).
    • In general,
      \[ fg(x) \neq gf(x) \]
    • \( ff(x) \) means you apply the function \( f \) twice.

2.3 Inverse functions

  • The inverse function is the opposite of the original function.
  • It is denoted by \( f^{-1}(x) \).
  • An inverse only exists for one-to-one functions.
  • Steps for finding the inverse function:
    • STEP-1: Write the function as \( y = f(x) \).
    • STEP-2: Interchange the \( x \) and \( y \) variables.
    • STEP-3: Rearrange to make \( y \) the subject.
  • The inverse function satisfies \( f(f^{-1}(x)) = f^{-1}(f(x)) = x \).
  • The domain of \( f^{-1}(x) \) is the range of \( f(x) \).
  • The range of \( f^{-1}(x) \) is the domain of \( f(x) \).
  • If \( f \) and \( f^{-1} \) are the same function, then \( f \) is called a self-inverse function.

2.4 The graph of a function and its inverse

  • The graphs of \( f \) and \( f^{-1} \) are reflections of each other in the line \( y = x \).
    • This is because \( f(f^{-1}(x)) = x = f^{-1}(f(x)) \).
  • When a function \( f \) is self-inverse, the graph of \( f \) will be symmetrical about the line \( y = x \).
  • Example:

2.5 Transformations of functions

Translations

  • In translation, the shape, size, and orientation of the graph remain unchanged, but the graph shifts up or down, left or right in the \( xy \) plane.
  • The graph of \( y = f(x) + a \) is a translation of the graph \( y = f(x) \) by the vector
    \[ \begin{bmatrix} 0 \\ a \end{bmatrix} \]
  • Example:
    • When the \( x \)-coordinates on the two graphs are the same (\( x = x \)), the \( y \)-coordinates differ by 3 (\( y = y + 3 \)).
    • This means that the two curves have exactly the same shape but are separated by 3 units in the positive \( y \)-direction.
    • Hence, the graph of \( y = x^2 – 2x + 4 \) is a translation of the graph of \( y = x^2 – 2x + 1 \) by the vector
      \[
      \begin{bmatrix} 0 \\ 3 \end{bmatrix}
      \]
  • The graph of \( y = f(x – a) \) is a translation of the graph \( y = f(x) \) by the vector
    \[
    \begin{bmatrix} a \\ 0 \end{bmatrix}
    \]
  • Example:
  • Equation of the red curve: \( y = x^2 – 2x + 1 \)
    Equation of the blue curve: \( y = (x – 3)^2 – 2(x – 3) + 1 \)
  • The curves have exactly the same shape but are separated by 3 units in the positive \( x \)-direction.
  • The \( y \)-coordinates are the same (\( y = y \)), but the \( x \)-coordinates differ by 3 (\( x = x + 3 \)).
  • This means that the two curves are at the same height when the blue curve is 3 units to the right of the red curve.
  • Hence, the graph of \( y = (x – 3)^2 – 2(x – 3) + 1 \) is a translation of the graph of \( y = x^2 – 2x + 1 \) by the vector
    \[
    \begin{bmatrix} 3 \\ 0 \end{bmatrix}
    \]
  • Combining these two results gives:
    The graph of \( y = f(x – a) + b \) is a translation of the graph \( y = f(x) \) by the vector
    \[
    \begin{bmatrix} a \\ b \end{bmatrix}
    \]

Reflections

  • The graph of \( y = -f(x) \) is a reflection of the graph \( y = f(x) \) in the \( x \)-axis.
  • Example:
  • The equation of the blue curve: \( y = x^2 – 2x + 1 \)
    The equation of the red curve: \( y = -(x^2 – 2x + 1) \)
  • When the \( x \)-coordinates on the two graphs are the same (\( x = x \)), the \( y \)-coordinates are negative of each other (\( y = -y \)).
  • This means that, when the \( x \)-coordinates are the same, the red curve is the same vertical distance from the \( x \)-axis as the blue curve but on the opposite side of the \( x \)-axis.
  • Hence, the graph of \( y = -(x^2 – 2x + 1) \) is a reflection of the graph of \( y = x^2 – 2x + 1 \) in the \( x \)-axis.
  • The graph of \( y = f(-x) \) is a reflection of the graph \( y = f(x) \) in the \( y \)-axis.
  • Example:

Stretches

  • The graph of \( y = a f(x) \) is a stretch of the graph \( y = f(x) \) with stretch factor \( a \) parallel to the \( y \)-axis.
  • Example:
  • The equation of the blue curve: \( y = x^2 – 2x – 3 \)
    The equation of the red curve: \( y = 2(x^2 – 2x – 3) \)
  • When the \( x \)-coordinates on the two graphs are the same \((x = x)\),
    the \( y \)-coordinate on the red graph is double the \( y \)-coordinate on the blue graph \((y = 2y)\).
  • This means that, when the \( x \)-coordinates are the same,
    the red curve is twice the distance of the blue graph from the \( x \)-axis.
  • Hence, the graph of \( y = 2(x^2 – 2x – 3) \) is a stretch of the graph of \( y = x^2 – 2x – 3 \)
    from the \( x \)-axis. We say that it has been stretched with stretch factor 2 parallel to the \( y \)-axis.
  • There are alternative ways of expressing this transformation:
    • A stretch with scale factor 2 with the line \( y = 0 \) invariant
    • A stretch with scale factor 2 with the \( x \)-axis invariant
    • A stretch with stretch factor 2 relative to the \( x \)-axis
    • A vertical stretch with stretch factor 2.
  • Note: if \( a < 0 \), then \( y = a f(x) \) can be considered to be a stretch of \( y = f(x) \)
    with a negative scale factor or as a stretch with positive scale factor followed by a reflection in the \( x \)-axis.
  • The graph of \( y = f(ax) \) is a stretch of the graph \( y = f(x) \)
    with stretch factor \( \frac{1}{a} \) parallel to the \( x \)-axis.
  • Example:
  • The equation of the blue curve: \( y = x^2 – 2x – 3 \)
    The equation of the red curve: \( y = (2x)^2 – 2(2x) – 3 \)
  • We obtain the second function by replacing \( x \) by \( 2x \) in the first function.
  • The two curves are at the same height \((y = y)\) when \( x = \frac{x}{2} \).
  • This means that the heights of the two graphs are the same
    when the red graph has half the horizontal displacement from the \( y \)-axis as the blue graph.
  • Hence, the graph of \( y = (2x)^2 – 2(2x) – 3 \) is a stretch of the graph of \( y = x^2 – 2x – 3 \)
    from the \( y \)-axis. We say that it has been stretched with stretch factor \( \frac{1}{2} \) parallel to the \( x \)-axis.

2.6 Combined transformations

  • When two vertical transformations or two horizontal transformations are combined, the order in which they are applied may affect the outcome.
  • When one horizontal and one vertical transformation are combined, the order in which they are applied does not affect the outcome.
  • Vertical transformation follows the ‘normal’ order of operations, as used in arithmetic. 
  • Horizontal transformations follow the opposite order to the ‘normal’ order of operations, as used in arithmetic.
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