Working with Functions

Working with functions is the foundation of the whole course, and it is the most heavily examined Year 11 focus area. In NESA's sample HSC paper for the 2024 syllabus it was worth 13 of the 100 marks, more than some Year 12 topics. It also turns up inside nearly every calculus, graphing and modelling question.

• Algebraic techniques: index laws (negative, zero and fractional powers), expanding and factorising, surds and rationalising denominators, solving quadratics three ways, the discriminant.
• Functions and relations: function notation, independent and dependent variables, the vertical line test, domain, range and zeros, interval notation.
• Linear functions: the forms of a straight line, parallel and perpendicular lines, linear inequalities.
• Quadratic and cubic functions: intercepts, axis of symmetry and vertex, finding a parabola's equation, equating coefficients, simultaneous equations, cubics like $y=k(x-a)(x-b)(x-c)$.
• Reciprocal functions: the hyperbola $y=\frac{k}{x}$ and its asymptotes.
• Direct and inverse variation: $y=k{x}^n$ and $y=\frac{k}{x^n}$.
• Circles and semicircles centred at the origin.
• Properties of functions: even and odd functions, composite functions.
• Piecewise and absolute value functions, including continuity.
• Modelling: using linear and quadratic functions to model a situation and justify a conclusion in context.

These skills are rarely examined on their own, but a slip here costs marks everywhere else.

To rationalise a denominator such as $\sqrt{5}-\sqrt{2}$, multiply top and bottom by its conjugate $\sqrt{5}+\sqrt{2}$, because $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=3$.

A quadratic $a{x}^2+bx+c=0$ can be solved by factorising, completing the square or the quadratic formula:

The discriminant $\Delta =b^2-4ac$ tells you how many real roots there are:

If $\Delta$ is a perfect square (and the coefficients are rational), the roots are rational.

Worked example: a tangent line using the discriminant

For what value of $k$ is the line $y=2x+k$ a tangent to the parabola $y=x^2-4x+10$?

A tangent meets the parabola exactly once, so set the equations equal and make $\Delta =0$.

The point of contact is the repeated root $x=3$, where $y=7$.

A relation pairs inputs with outputs. A function is a relation where every input ($x$-value) has exactly one output. On a graph, use the vertical line test: if any vertical line cuts the graph more than once, it is not a function. A circle is a relation but not a function.

• The domain is the set of allowed $x$-values; the range is the set of resulting $y$-values.
• The natural domain excludes anything that breaks the rule: no zero denominators, nothing negative under an even root.
• The zeros of $f$ are the solutions of $f(x)=0$, which are the $x$-intercepts of the graph.

Write domains and ranges in interval notation, inequalities or words. Square brackets include an endpoint, round brackets exclude it: $[2,5)$ means $2\le x<5$, and $(-\infty ,3]$ means $x\le 3$. Infinity always takes a round bracket. Use $\cup$ to join two intervals.

Composite functions

$f(g(x))$ means apply $g$ first, then $f$. In general $f(g(x))\ne g(f(x))$. The domain of $f(g(x))$ is every $x$ in the domain of $g$ for which $g(x)$ lies in the domain of $f$.

Example. Let $f(x)=\sqrt{x+1}$ and $g(x)=x^2-5$.

We need $x^2-4\ge 0$, so the domain is $x\le -2$ or $x\ge 2$, that is $(-\infty ,-2]\cup [2,\infty )$. The range is $[0,\infty )$.

Even and odd functions

• Even: $f(-x)=f(x)$ for all $x$. The graph is symmetric about the $y$-axis, for example $x^2$, $x^4-3$, $|x|$.
• Odd: $f(-x)=-f(x)$ for all $x$. The graph has point symmetry about the origin, for example $x^3$, $\frac{1}{x}$, $x^3-2x$.
• Most functions are neither. To show it, find one value where both tests fail.

Always test algebraically: substitute $-x$, simplify, and compare with $f(x)$ and $-f(x)$.

Straight lines

Parallel lines have equal gradients. Perpendicular lines have $m_1{m}_2=-1$. When you solve a linear inequality, reverse the inequality sign if you multiply or divide by a negative number.

Parabolas

For $y=a{x}^2+bx+c$, the axis of symmetry is $x=-\frac{b}{2a}$, and the vertex lies on it. Completing the square gives the vertex form $y=a(x-h{)}^2+k$ with vertex $(h,k)$. If $a>0$ the parabola is concave up and the vertex is a minimum.

To find a parabola from its graph, use the form that matches the information: $y=a(x-\alpha )(x-\beta )$ when you know the $x$-intercepts, or $y=a(x-h{)}^2+k$ when you know the vertex. Then substitute one more point to find $a$.

Equating coefficients: if two quadratics are equal for all $x$, their $x^2$ coefficients, $x$ coefficients and constants are each equal.

Cubics

Graph $y=k{x}^3$ (a point of inflection at the origin) and $y=k(x-a)(x-b)(x-c)$ (three $x$-intercepts). For $k>0$ the graph rises from bottom left to top right; for $k<0$ it falls. A squared factor such as $(x-a{)}^2$ means the graph touches the axis at $x=a$ instead of crossing it.

Hyperbolas

$y=\frac{k}{x}$ has asymptotes $x=0$ and $y=0$. For $k>0$ the branches are in the first and third quadrants; for $k<0$ they are in the second and fourth. As $x\to \pm \infty$, $y\to 0$.

Simultaneous equations and graphs

The solutions of $f(x)=g(x)$ are the $x$-coordinates of the intersection points of the two graphs. Solve a line and a parabola, or two parabolas, by substitution. The number of intersections is often asked; sketch first, then confirm algebraically.

• Direct variation: $y=k{x}^n$, read "$y$ varies directly as $x^n$".
• Inverse variation: $y=\frac{k}{x^n}$, read "$y$ varies inversely as $x^n$".

The method is always the same: write the equation with $k$, use one pair of values to find $k$, then use the equation.

Worked example: braking distance

A car's braking distance $d$ metres varies directly as the square of its speed $v$ km/h. At 60 km/h the braking distance is 18 m. Find the braking distance at 90 km/h, and the speed that gives a braking distance of 50 m.

Notice that multiplying the speed by 1.5 multiplied the distance by $1.5^2=2.25$. Reject the negative square root because speed is positive here, and say so.

By Pythagoras' theorem, a point $(x,y)$ at distance $r$ from the origin satisfies

Solving for $y$ gives two semicircles: $y=\sqrt{r^2-x^2}$ is the upper half (range $[0,r]$) and $y=-\sqrt{r^2-x^2}$ is the lower half. Each semicircle is a function; the whole circle is not. Both have domain $[-r,r]$. Circles with other centres belong to Graph transformations.

A piecewise function uses different rules on different parts of the domain. To evaluate it, first decide which piece the $x$-value belongs to; watch whether each boundary uses $<$ or $\le$. When you graph one, use a closed dot for an included endpoint and an open dot for an excluded one.

A function is continuous at a point if its graph has no break there. For a piecewise function, check each joining point: the pieces must meet at the same $y$-value. If they don't, there is a jump and the function is discontinuous there.

The absolute value $|x|$ is the distance of $x$ from 0:

Also $\sqrt{x^2}=|x|$, not $x$. The graph of $y=a|x|+b$ is a V shape with its vertex at $(0,b)$, opening up if $a>0$.

Worked example: an absolute value equation

Solve $|x+1|=2x-4$.

The left side is never negative, so any solution needs $2x-4\ge 0$, that is $x\ge 2$. Now solve both cases.

$x=1$ fails $x\ge 2$ (check: $|2|=2$ but $2(1)-4=-2$), so it is rejected. The only solution is $x=5$. Graphically, the line $y=2x-4$ meets the V of $y=|x+1|$ only once.

• Writing the domain of $\frac{1}{x-3}$ as $x>3$. Only $x=3$ is excluded.
• Using a square bracket next to $\infty$, or forgetting the $\cup$ between two intervals.
• Mixing up $f(g(x))$ and $g(f(x))$. Work from the inside out.
• Claiming a function is even or odd from one value. You need $f(-x)=\pm f(x)$ for every $x$.
• Forgetting to reverse the inequality when dividing by a negative.
• Using $m_1=-m_2$ instead of $m_1{m}_2=-1$ for perpendicular lines.
• Not checking absolute value solutions in the original equation.
• In variation questions, using $y=kx$ when the question says "the square of $x$".

• Show that and prove questions need every line of working; you cannot jump to the given answer.
• For sketch questions, label intercepts, the vertex or turning points, asymptotes and endpoints (open or closed dots). A sketch does not need to be to scale but must show these features.
• In modelling questions, finish with a sentence in context (units, what the number means) and state any restriction, such as rejecting a negative time or length.
• Discriminant questions usually turn on the right condition: $\Delta >0$, $\Delta =0$, $\Delta \ge 0$ or $\Delta <0$. Write the condition before you solve.
• Domain and range answers are often worth a mark each. Check for excluded points and for whether endpoints are included.

• Graph transformations (Year 11): shifting, stretching and reflecting the functions in this topic, and circles with any centre.
• Introduction to differentiation: tangents, normals and stationary points all rely on lines, quadratics and cubics.
• Further graph transformations and modelling (Year 12): solving equations graphically and modelling with functions.
• Applications of calculus: optimisation and curve sketching start with domain, intercepts and the shape of the function.