Further Work with Functions

functions year11 ext1

Further work with functions | NSW Curriculum Website

• ME1-11-01 solves problems involving inequalities, functions and their inverses, graphical relationships between functions, and parametric equations

📖 Prior Knowledge

Content	Prerequisite relationships
Equations B	- Solve linear inequalities → Solve absolute value inequalities
Equations C	- Change the subject of a formula → Find the inverse of a function
Non-Linear Relationships C	- Sketch parabolas from their equation → Graph curves in parametric form - Graph circles centred at the origin → Express relations in parametric form - Identify graph types from their equations → Graph the reciprocal of a function - Identify graph types from their equations → Graph sums and differences of functions - Find intersections of a line and a curve → Solve inverse function problems
Polynomials	- Apply the factor theorem → Solve cubic inequalities
Functions and other Graphs	- Use function notation f(x) → Identify one-to-one functions - Use function notation f(x) → Understand parametric equations
Working with Functions	- Solve quadratic inequalities → Solve cubic inequalities - Solve quadratic inequalities → Solve rational inequalities - Describe the behaviour of a reciprocal function → Graph the reciprocal of a function - Express domain and range in interval notation → Domain and range of sums and differences - Express domain and range in interval notation → Identify one-to-one functions - Express domain and range in interval notation → Domain and range of an inverse - Form and evaluate composite functions → Verify inverses by composition - Graph absolute value functions → Graph $y=|f(x)|$ and $y=f(|x|)$ - Solve absolute value equations → Solve absolute value inequalities
Trigonometry and Measure of Angles	- Graph trig functions in radians → Graph reciprocal trigonometric functions
Trigonometric Identities and Equations	- Define sec, cosec and cot → Graph reciprocal trigonometric functions
Graph Transformations	- Reflect graphs in the axes → Reflect in the line $y=x$ - Apply combined transformations → Graph $y=|f(x)|$ and $y=f(|x|)$

Graphical relationships

• Examine the relationship between the graph of $y=f(x)$ and the graph of $y=\frac{1}{f(x)}$ using graphing applications, and graph $y=\frac{1}{f(x)}$ given $y=f(x)$ in algebraic or graphical form, identifying any vertical and horizontal asymptotes of $y=f(x)$ and $y=\frac{1}{f(x)}$
• Graph $y=\sec x$, $y=\mathrm{cosec}x$ and $y=\cot x$ in both radians and degrees, identifying key properties including asymptotes, period, domain, range and symmetry, and compare each graph with the graph of its reciprocal
• Examine the relationship between the graph of $y=f(x)$ and the graphs of $y=|f(x)|$ and $y=f(|x|)$ using graphing applications, and graph $y=|f(x)|$ and $y=f(|x|)$ given $y=f(x)$ in algebraic or graphical form
• Examine the relationship between the graphs of $y=f(x)$ and $y=g(x)$ and the graphs of $y=f(x)+g(x)$ and $y=f(x)-g(x)$ using graphing applications, and graph $y=f(x)+g(x)$ and $y=f(x)-g(x)$ given $y=f(x)$ and $y=g(x)$ in algebraic or graphical form
• Determine the domains and ranges of the sum and difference of functions where possible, and, if appropriate, verify them using a graphing application
• Apply knowledge of graphical relationships to solve problems involving graphs of functions, justifying conclusions in the context of the problem where appropriate

Inverse functions

• Describe a function as one-to-one if every element in the range of the function corresponds to exactly one element of the domain
• Establish that the reflection of a point in the line $y=x$ reverses the coordinates of the point
• Define an inverse function $f^{-1}$ informally as a function that reverses or undoes the effect of the function $f$
• Recognise that inverse functions exist for one-to-one functions
• Determine the equation for the inverse function $f^{-1}(x)$ of a given one-to-one function $y=f(x)$ by interchanging the variables $x$ and $y$ in $y=f(x)$
• Compare the graphs of a function and its inverse function using graphing applications, and recognise that the two graphs are reflections of each other in the line $y=x$
• Establish that the domain of $f^{-1}(x)$ is the range of $f(x)$, and the range of $f^{-1}(x)$ is the domain of $f(x)$, and use this relationship to solve problems
• Graph the inverse function of a given one-to-one function
• Explain that the reflection in the line $y=x$ exchanges horizontal and vertical lines and recognise that the horizontal line test can therefore be applied to the graph of $y=f(x)$ to determine whether its reflection in the line $y=x$ is a function
• Apply restrictions to the domain of a function, if it is not one-to-one, to obtain an inverse function
• Define formally $f^{-1}(x)$ to be the inverse function of $f(x)$ if the relationships $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$ hold, and use this definition to solve problems
• Solve problems based on the relationship between a function and its inverse function using algebraic and graphical techniques, including determining the points of intersection of a function and its inverse, where they exist

Parametric form of a function or relation

• Recognise that a curve may be represented by two parametric equations that give $x$ and $y$ as functions of a parameter and that the curve may also have a Cartesian equation in $x$ and $y$
• Express linear and quadratic functions and circles in parametric form
• Convert linear and quadratic functions and circles from parametric form to Cartesian form
• Graph linear and quadratic functions and circles expressed in parametric form

Inequalities

• Solve cubic inequalities where the cubic is expressed as a product of linear factors
• Solve inequalities involving rational expressions with variables in the denominator
• Solve absolute value inequalities of the form $|ax+b|\ge k$, $|ax+b|\le k$, $|ax+b|<k$, $|ax+b|>k$, where $a$, $b$ and $k$ are constants, using algebraic and graphical methods or the characterisation of $|x|$ as the distance of $x$ from the origin