Extension Functions – Working with functions

This lesson comprises six (6) master classes focusing on:

• Reciprocal functions
• Inverse functions
• Parametric representation
• Inequalities

Content:

ME-F1.1

• Examine the relationship between the graph of \( y=f(x) \) and the graph of \( y=\frac{1}{f(x)} \) and hence sketch the graphs
• Examine the relationship between the graph of \( y=f(x) \) and the graphs of \( y^2=f(x) \) and \( y=\sqrt{f(x)} \) and hence sketch the graphs
• Examine the relationship between the graph of \( y=f(x) \) and the graphs of \( y=|f(x)| \) and \( y=f(|x|) \) and hence sketch the graphs
• Examine the relationship between the graphs of \( y=f(x) \) and \( y=g(x) \) and the graphs of \( y=f(x)+g(x) \) and \( y=f(x)g(x) \) and hence sketch the graphs
• Apply knowledge of graphical relationships to solve problems in practical and abstract contexts

ME-F1.2

• Solve quadratic inequalities using both algebraic and graphical techniques
• Solve inequalities involving rational expressions, including those with the unknown in the denominator
• Solve absolute value inequalities of the form \( |ax+b| \ge k \), \(|ax+b| \le k \), \( |ax+b|<k \) and \( |ax+b|>k \)

ME-F1.3

• Define the inverse relation of a function \( y=f(x) \) to be the relation obtained by reversing all the ordered pairs of the function
• Examine and use the reflection property of the graph of a function and the graph of its inverse
  • understand why the graph of the inverse relation is obtained by reflecting the graph of the function in the line \( y=x \)
  • using the fact that this reflection exchanges horizontal and vertical lines, recognise that the horizontal line test can be used to determine whether the inverse relation of a function is again a function
• Write the rule or rules for the inverse relation by exchanging x and y in the function rules, including any restrictions, and solve for y, if possible
• When the inverse relation is a function, use the notation \( f^{−1}(x) \) and identify the relationships between the domains and ranges of \( f(x) \) and \( f^{−1}(x) \)
• When the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions
• Solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques

ME-F1.4

• Understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form
  • understand that linear and quadratic functions, and circles can be expressed in either parametric form or Cartesian form
  • convert linear and quadratic functions, and circles from parametric form to Cartesian form and vice versa
  • sketch linear and quadratic functions, and circles expressed in parametric form