Working with Functions

functions year11 advanced

Working with functions | NSW Curriculum Website

• MAV-11-01 applies algebraic techniques and the laws of indices and surds to manipulate expressions and solve problems
• MAV-11-02 uses functions and relations to model, analyse and solve problems

📖 Prior Knowledge

Content	Prerequisite relationships
Represents Numbers A	- Approximate decimals to a given number of places → Solve real-life linear relationship problems
Computation with Integers	- Determine the sign of a product or quotient → Expand brackets using the distributive law
Algebraic Techniques	- Substitute values into algebraic expressions → Use function notation f(x) - Substitute values into algebraic expressions → Model direct variation $y=k{x}^n$ - Substitute values into algebraic expressions → Model inverse variation $y=k/x^n$ - Multiply and divide algebraic terms → Expand brackets using the distributive law - Multiply and divide algebraic terms → Apply index laws to algebraic terms - Expand a bracket with a negative term in front → Expand brackets using the distributive law - Expand brackets using the distributive law (revisited content)
Indices	- Use square roots and cube roots → Use $\sqrt{x^2}=|x|$ - Apply index laws with numerical bases → Apply index laws to algebraic terms
Equations	- Solve 2-step linear equations → Solve real-life linear relationship problems - Verify equation solutions by substitution → Model and solve word problems with quadratics
Linear Relationships	- Plot points on the Cartesian plane → Define relations and functions - Graph a linear relationship → Solve real-life linear relationship problems - Graph a linear relationship → Graph absolute value functions - Solve real-life linear relationship problems (revisited content)
Algebraic Techniques A	- Expand binomial products → Expand and simplify expressions involving surds
Indices A	- Apply index laws to algebraic terms (revisited content)
Equations A	- Solve 3-step linear equations → Solve linear inequalities - Solve 3-step linear equations → Solve simultaneous equations algebraically
Linear Relationships A	- Find the gradient of an interval → Recognise and graph direct variation
Linear Relationships B	- Interpret gradient–intercept form → Find a line’s equation from gradient and intercept - Interpret gradient–intercept form → Convert between gradient–intercept and general form - Find a line’s equation from gradient and intercept (revisited content) - Find equations of parallel and perpendicular lines → Find and test parallel and perpendicular lines in any form
Variation and Rates of Change A	- Describe and represent direct variation → Recognise and graph direct variation - Describe and represent inverse variation → Recognise inverse variation from graphs - Recognise and graph direct variation (revisited content) - Recognise inverse variation from graphs (revisited content)
Algebraic Techniques B	- Operate with algebraic fractions with pronumerals in the denominator → Simplify algebraic fractions by factorising - Factorise monic quadratic trinomials → Factorise using special products and grouping - Factorise monic quadratic trinomials → Find intercepts, axis of symmetry and vertex of a parabola
Algebraic Techniques C	- Prove and apply special products → Factorise using special products and grouping - Prove and apply special products → Complete the square - Factorise using special products and grouping (revisited content) - Simplify algebraic fractions by factorising (revisited content)
Indices B	- Apply index laws with negative-integer indices → Describe and use fractional indices
Indices C	- Determine when square roots are defined → Graph the square root function - Apply the basic surd results → Simplify expressions involving surds - Apply the basic surd results → Describe and use fractional indices - Simplify expressions involving surds (revisited content) - Expand and simplify expressions involving surds (revisited content) - Rationalise the denominator of a surd (revisited content) - Describe and use fractional indices (revisited content) - Rationalise binomial surd denominators (revisited content)
Equations B	- Solve monic quadratic equations by factorising → Solve quadratic equations by factorisation - Represent inequalities on a number line → Solve linear inequalities - Solve linear inequalities (revisited content)
Equations C	- Change the subject of a formula → Convert between gradient–intercept and general form - Solve quadratic equations by factorisation (revisited content) - Complete the square (revisited content) - Solve quadratics by completing the square (revisited content) - Apply the quadratic formula (revisited content) - Use the discriminant to determine the nature of roots (revisited content) - Find unknown coefficients using the discriminant (revisited content) - Model and solve word problems with quadratics (revisited content) - Solve simultaneous equations algebraically (revisited content) - Model word problems with simultaneous equations (revisited content) - Find unknowns by equating coefficients → Equate coefficients of equal quadratics
Linear Relationships C	- Apply the gradient formula → Find a line’s equation using point–gradient form - Apply the distance formula → Graph circles centred at the origin - Convert between gradient–intercept and general form (revisited content) - Find a line’s equation using point–gradient form (revisited content) - Find and test parallel and perpendicular lines in any form (revisited content)
Non-Linear Relationships C	- Graph parabolas and describe transformations → Find intercepts, axis of symmetry and vertex of a parabola - Graph parabolas and describe transformations → Graph and compare power curves - Find intercepts, axis of symmetry and vertex of a parabola (revisited content) - Sketch parabolas from their equation (revisited content) - Graph hyperbolas and describe transformations (revisited content) - Graph circles centred at the origin (revisited content) - Find intersections of a line and a curve (revisited content) - Graph and compare power curves (revisited content) - Find the equation of a parabola from features (revisited content)
Polynomials	- Apply the factor theorem → Graph polynomials from factored form - Graph polynomials from factored form (revisited content)
Functions and other Graphs	- Define relations and functions (revisited content) - Apply the vertical line test (revisited content) - Use function notation f(x) (revisited content) - Find the domain and range of a function (revisited content) - Read and write interval notation → Find the domain and range of a function
Algebraic Relationships	- Analyse break-even points (revisited content)

Algebraic techniques

• Use index laws to simplify expressions and solve problems involving positive, negative, zero or fractional indices
• Expand, factorise and simplify algebraic expressions
• Simplify expressions involving algebraic fractions
• Expand and simplify expressions involving surds
• Identify the conjugate of $\sqrt{a}\pm \sqrt{b}$, and rationalise the denominators of expressions of the form $\frac{a}{\sqrt{b}\pm \sqrt{c}}$ and $(\sqrt{a}\pm \sqrt{d})/(\sqrt{b}\pm \sqrt{c})$, where $a,b,c$ and $d$ are positive rational numbers
• Solve quadratic equations $a{x}^2+bx+c=0$ by factorisation, completing the square and using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ where $a,b$ and $c$ are real numbers and $a\ne 0$
• Define the discriminant as $\Delta =b^2-4ac$ and use it to solve problems involving the number of real roots of a quadratic equation and determine the conditions for roots to be equal, distinct, real or rational

Introduction to functions and relations

• Describe a relation between two sets as an association between the elements of one set and the elements of the other set
• Define a real function $f$ of a real variable $x$ as a relation where each element $x$ of a given set is associated with exactly one element $y$ of the second set
• Recognise that a relation is a rule which can be represented by an algebraic formula, a table of values, a set of ordered pairs $(x,y)$ or a graph
• Use the notation$f(x)$ to identify the unique value of $y$ associated with $x$ when working with functions, and refer to it as the value of $f$ at $x$
• Refer to $x$ in a function $y=f(x)$ as the independent variable of the function, and refer to $y$ as the dependent variable
• Substitute numeric and algebraic expressions into the formulas of functions
• Apply the vertical line test on the graph of a relation to determine whether it represents a function
• Define the domain of a function $f$ as the set of real numbers on which $f$ is defined
• Define the range of a function $f$ as the set of values of $f(x)$ obtained as $x$ varies over the domain of $f$
• Refer to a value in the domain of a function at which the function is 0 as a zero of the function
• Recognise that the $x$-intercepts of the graph of $y=f(x)$ are its zeroes, the solutions of $f\left(x\right)=0$, and that the $y$-intercept is $f(0)$

Linear functions

• Determine the equations of straight lines in gradient–intercept form $y=mx+c$ with gradient $m$ and $y$-intercept $c$
• Determine the equations of straight lines in general form $ax+by+c=0$, where $a$, $b$ and $c$ are constants
• Determine the equation of a straight line passing through a point $(x,y)$ with gradient $m$ using the point–gradient formula $y-y_1=m\left(x-x_1\right)$
• Determine the equation of a straight line passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ by calculating its gradient m using the formula $m=\frac{y_2-y_1}{x_2-x_1}$
• Determine the $x$-intercept and $y$-intercept of a straight line given its equation
• Choose and apply appropriate techniques to graph a straight line given its equation
• Find the equation of a line that is parallel or perpendicular to a given line
• Solve linear inequalities and graph the solution on a number line

Quadratic and cubic functions

• Identify the $x$-intercepts of a parabola whose quadratic function is expressed in factored form
• Use the discriminant to determine the number of $x$-intercepts on a parabola and justify its position in relation to the $x$-axis
• Show by completing the square on the general quadratic $y=a{x}^2+bx+c$ that the axis of symmetry is $x=-\frac{b}{2a}$ where $a,b$ and $c$ are constants
• Identify the axis of symmetry and vertex of a parabola by completing the square on its quadratic function
• Choose and apply appropriate techniques to graph a parabola of the form $y={ax}^2+bx+c$ by identifying its $x$-intercepts if they exist, its $y$-intercept, its axis of symmetry using $x=-\frac{b}{2a}$ and its vertex
• Find the equation of a parabola given sufficient graphical features
• Use the fact that two quadratic functions are equal for all values of x if and only if the corresponding coefficients are equal to solve related problems
• Recognise that solving $f\left(x\right)=k$ for some constant $k$ corresponds to finding the $x$-coordinate(s) of the intersection of the graphs $y=f\left(x\right)$ and $y=k$
• Solve problems by finding the solution to simultaneous equations involving a linear and a quadratic function, or two quadratic functions, both algebraically and graphically
• Solve quadratic inequalities
• Recognise and graph cubic functions of the form $f(x)=k{x}^3$ and $f\left(x\right)=k\left(x-a\right)\left(x-b\right)\left(x-c\right)$, where $a,b,c$ and $k$ are constants and $k\ne 0$

Reciprocal functions

• Graph functions of the form $f(x)=\frac{k}{x}$, where $k$ is a constant and $k\ne 0$, and identify their hyperbolic shape and their asymptotes
• Describe the behaviour of $f\left(x\right)=\frac{k}{x}$ as $x\to \infty$ and $x\to -\infty$

Constructing and using functions

• Construct and use linear functions to model and solve problems in real-world situations, identifying the independent and dependent variables and any restrictions on these variables, and justify conclusions in the context of the problem
• Use linear inequalities to model and solve problems in real-world situations, and justify conclusions in the context of the problem
• Solve practical problems involving a pair of simultaneous linear equations both algebraically and graphically, with and without graphing applications, and justify conclusions in the context of the problem
• Construct and use simultaneous equations to model and solve a problem where cost and revenue are represented by linear equations, identify and analyse the break-even point, and justify conclusions in the context of the problem
• Model and solve practical problems involving quadratic functions and justify conclusions in the context of the problem

Direct and inverse variation

• Develop models of the form $y=k{x}^n$, where k is a non-zero constant, from descriptions of situations in which one quantity varies directly with another
• Develop the model $y=\frac{k}{x^n}$, where k is a non-zero constant, from descriptions of situations in which one quantity varies inversely with another
• Evaluate $k$ in the equations $y=k{x}^n$ and $y=\frac{k}{x^n}$, given one pair of values for the variables, and use the resulting formula to find other values of the variables
• Analyse and solve problems involving direct and inverse variation

Circles and semicircles

• Derive the equation of a circle of radius $r$ with centre at the origin by considering Pythagoras’ theorem
• Graph circles of the form $x^2+y^2=r^2$ from their equations
• Determine the equation of a circle of the form $x^2+y^2=r^2$ given its graph
• Identify and graph the semicircles $y=\sqrt{r^2-x^2}$, $y=-\sqrt{r^2-x^2}$, $x=\sqrt{r^2-y^2}$ and $x=-\sqrt{r^2-y^2}$

Properties of functions, relations and graphs

• Extend the definitions of domain and range to relations
• Recognise domains and ranges of functions and relations given in interval notation, as inequalities and as worded descriptions
• Determine and describe the domain and range of functions and relations, using interval notation, inequalities or worded descriptions
• Define a function to be even if its graph is unchanged under reflection in the y-axis, and odd if its graph is unchanged under rotation of 180° about the origin
• Develop and use the tests that a function $f(x)$ is odd if $f\left(-x\right)=-f(x)$ and a function $f(x)$ is even if $f\left(-x\right)=f(x)$
• Solve problems involving even and odd functions
• Use the composite function $f\left(g\left(x\right)\right)$, where the output of $g\left(x\right)$ becomes the input of $f\left(x\right)$
• Determine the equations of composite functions

Piecewise-defined functions

• Interpret piecewise-defined functions, where the function is defined differently in different parts of the domain
• Graph piecewise-defined functions involving functions covered in the scope of the Mathematics Advanced course, test if they are even or odd, and determine the domain and range
• Define informally that a function is continuous at a point if the curve can be drawn through the point without lifting the pen off the paper
• Identify points where piecewise-defined functions and other functions are not continuous
• Define a discontinuity of a function informally as a point where the function is not continuous

Absolute value functions

• Define the absolute value $\left|x\right|$ of a number $x$, also known as the magnitude of $x$, to be the distance from the origin to x on the number line
• Establish and use the piecewise definition $|x|=\{\begin{array}{ll}x,& \mathrm{for}x\ge 0,\\ -x,& \mathrm{for}x<0\end{array}$
• Show using numerical substitutions that $\sqrt{x^2}=\left|x\right|$ and use the result
• Graph the function $y=\left|x\right|$, describe its symmetry, and identify its domain and range
• Graph $y=\left|ax+b\right|$ with and without graphing applications, and identify its symmetry, domain and range
• Solve absolute value equations of the form $\left|ax+b\right|=k$ algebraically and graphically