overtime assesment y13 pure

• What is a mapping in the context of functions?

  A mapping is a function if every x-value maps to a single y-value.
• What is the image of x in a function?

  The output value corresponding to the input x is called the image of x.
• What defines a one-to-one function?

  A function is one-to-one if every y-value corresponds to only one x-value.
• What tests can be applied to graphs of mappings?

  The vertical and horizontal line tests can be applied to graphs of mappings.
• What does the vertical line test determine?

  If a mapping is a function, any vertical line will cross its graph at most once.
• What defines a many-to-one function?

  A function is many-to-one if there are some y-values that come from more than one x-value.
• What is the domain of a function?

  The set of allowed input values of a function is called the domain.
• What does the horizontal line test determine?

  If a function is one-to-one, any horizontal line will cross the graph at most once.
• What is the range of a function?

  The set of all possible outputs of a function is called the range.
• How is a composite function written?

  The composite function formed by applying g to x and then f to the result is written as f(g(x)) or fg(x) or fog(x).
• What is the inverse of a function f?

  The inverse, f', of a function f is such that ff'(x)=ff(x)=x, for all x.
• What are the steps to find the inverse function f'(x) given f(x)?

  1. Start with \( y = f(x) \)
  2. Rearrange to get \( x \) in terms of \( y \)
  3. Replace the \( y \)s with \( x \)s to get \( f'(x) \)
• Which functions have inverse functions?

  Only one-to-one functions have inverse functions.
• How does the graph of \( y=f(x) \) relate to its inverse?

  The graph of \( y=f(x) \) is a reflection of the graph of \( y = f^{-1}(x) \) in the line \( y = x \).
• What is the relationship between the domain and range of a function and its inverse?

  The domain of \( f(x) \) is the same as the range of \( f^{-1}(x) \).
• What is the relationship between the range and domain of a function and its inverse?

  The range of \( f(x) \) is the same as the domain of \( f^{-1}(x) \).
• How do you solve the equation \( |x + k| = |x| \) where \( k > 0 \)?

  Consider the cases for \( x + k \) and \( x \) based on their signs.
• What transformations can be combined on graphs?

  • Translations
  • Stretches
  • Reflections (horizontal and vertical)
• How does the order of transformations affect the outcome?

  The order in which transformations occur may affect the outcome.
• Can one horizontal and one vertical transformation be done in either order?

  Yes, one horizontal and one vertical transformation can be done in either order.
• How does changing the order of two horizontal or two vertical transformations affect the outcome?

  Changing the order of two horizontal or two vertical transformations affects the outcome.
• In the function \( y = pf(x) + c \), which transformation is performed first?

  The stretch is performed before the translation.
• In the function \( y = f(qx + d) \), which transformation is performed first?

  The translation is performed before the stretch.
• How can the modulus function be used in graph transformations?

  The modulus function can be used to reflect the part of the graph below the x-axis so that the whole graph is above it.
• How should equations and inequalities involving the modulus function be solved?

  Always use graphs to decide whether the solutions are on the reflected or unreflected part of the graph.
• How can a sequence be defined?

  A sequence can be defined by a formula for the nth term or by a terms-to-term rule.
• What characterizes an increasing sequence?

  An increasing sequence is characterized by each term being larger than the previous one: \( u_{n+1} > u_n \).
• What characterizes a decreasing sequence?

  A decreasing sequence is characterized by each term being smaller than the previous one: \( u_{n+1} < u_n \).
• What characterizes a periodic sequence?

  A periodic sequence is characterized by terms starting to repeat after a while: \( u_{n+k} = u_n \) for some number \( k \) (the period of the sequence).
• What defines an arithmetic sequence?

  An arithmetic sequence has a constant difference, \( d \), between consecutive terms.
• What is a series?

  • A series is a sum of terms in a sequence.
  • It can be described in shorthand using sigma notation:

  \(\sum_{i=1}^n f(i) = f(1) + f(2) + ... + f(n)\)
• What is the formula for the nth term of an arithmetic sequence?

  If you know the first term, \( a \), the nth term is: \( u_n = a + (n - 1)d \).
• What is the formula for the sum of the first n terms in an arithmetic sequence?

  If you know the first term and the common difference, the sum of all n terms is: \( S_n = \frac{n}{2} [2a + (n - 1)d] \).
• What is the formula for the sum of the first n terms in an arithmetic sequence if you know the first and last term?

  If you know the first and last term \( (l) \), the sum of all n terms is: \( S_n = \frac{n}{2}(a + l) \).
• What defines a geometric sequence?

  A geometric sequence has a constant ratio, \( r \), between consecutive terms.
• What is the formula for the nth term of a geometric sequence?

  If you know the first term, \( a \), the nth term is: \( u_n = ar^{n-1} \).
• What is the formula for the sum of the first n terms in a geometric sequence?

  The sum of the first n terms is: \( S_n = \frac{a(1 - r^n)}{1 - r} \) or equivalently: \( S_n = \frac{a(r^n - 1)}{r - 1} \).
• Under what condition does a geometric series converge?

  If \( |r| < 1 \), the series converges and the sum to infinity is given by: \( S_\infty = \frac{a}{1 - r} \).