← Maths Extension 1 syllabus

NSWMaths Extension 1

Functions (ME-F1)

7 dot points across 7 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we solve an inequation that contains an absolute value, or that has the unknown in the denominator, without losing or inventing solutions?

Solve absolute value inequations of the form |ax + b| < k, and inequations with the unknown in the denominator, by multiplying through by the square of the denominator
The Year 11 Extension 1 dot point on inequations with an absolute value or a pronumeral in the denominator. Three equivalent methods for |ax + b| < k, the multiply-by-the-square trick for rational inequations, the degenerate cases, and where students lose marks.
16 min answer →

Given the graph of $y = f(x)$ , how do we sketch the graph of its reciprocal $y = 1/f(x)$ without first finding a formula?

Sketch the graph of $y = \dfrac{1}{f(x)}$ from the graph of $y = f(x)$ : turn zeroes into vertical asymptotes, send large values to small ones, fix the points where $f = \pm 1$ , and flip a maximum to a minimum where $f$ keeps its sign
The Year 11 Extension 1 dot point on sketching the reciprocal of a graphed function. Why zeroes of f become vertical asymptotes of 1/f, why large values become small, why the points where f equals plus or minus one are fixed, why a maximum flips to a minimum where f keeps its sign, how a horizontal asymptote at y = L moves to y = 1/L, and the edge cases textbooks bury.
18 min answer →

Given the graphs of $y = f(x)$ and $y = g(x)$ , how do we sketch their sum and difference without first finding a formula?

Sketch the graph of $y = f(x) \pm g(x)$ by adding or subtracting ordinates: at a zero of one function the sum meets the other, opposite ordinates give a zero, equal ordinates double, and an oblique asymptote can emerge
The Year 11 Extension 1 dot point on sketching the sum or difference of two graphed functions by adding ordinates. Why a zero of one curve makes the sum meet the other, why opposite ordinates give a zero, why equal ordinates double, how an oblique asymptote emerges from a rational plus a line, why a sum of straight pieces stays straight, and the symmetry and domain rules textbooks rush.
18 min answer →

Given the graph of $y = f(x)$ , how do we sketch $y = |f(x)|$ and $y = f(|x|)$ , and why are they two completely different transformations?

Sketch $y = |f(x)|$ by reflecting the part of $y = f(x)$ below the $x$ -axis upward, and sketch $y = f(|x|)$ by keeping the part right of the $y$ -axis and mirroring it across the $y$ -axis, recognising that $y = f(|x|)$ is always even and that the two transformations coincide only in special cases
The Year 11 Extension 1 dot point on the two absolute-value graph transformations. Why y = |f(x)| flips the part below the x-axis upward (so y stays at least zero) while y = f(|x|) discards the left half and mirrors the right half (always even), where they differ, how to combine them in order, and why |2^x| = 2^x.
20 min answer →

How do we form the inverse of a relation by swapping $x$ and $y$ , when is that inverse itself a function, and how do we find and verify a rule for $f^{-1}(x)$ ?

Form the inverse relation by reflecting $y = f(x)$ in the line $y = x$ , use the horizontal line test to decide whether the inverse is a function, find the rule for $f^{-1}(x)$ and verify it by showing $f(f^{-1}(x)) = x$ , swap the domain and range, and restrict a domain so that a many-to-one function gains an inverse
The Year 11 Extension 1 dot point on inverse relations and functions. Swap x and y to get the inverse (a reflection in y = x), use the horizontal line test to decide whether it is a function, find and verify f-inverse by composition, swap the domain and range, and restrict a domain so a many-to-one function gains an inverse.
22 min answer →

How does a pair of equations $x = f(t)$ and $y = g(t)$ describe a curve, how do we eliminate the parameter $t$ to recover the Cartesian equation, and how do we read off the direction of travel and any excluded points?

Define a curve parametrically by $x = f(t)$ and $y = g(t)$ , eliminate the parameter $t$ by algebra or by a Pythagorean identity to obtain the Cartesian equation, parametrise standard curves (lines, parabolas, circles, ellipses and the rectangular hyperbola), and determine the direction of travel (orientation) and any excluded points
The Year 11 Extension 1 dot point on parametric equations. Write a curve as x = f(t) and y = g(t), eliminate the parameter t by algebra or the Pythagorean identity to get the Cartesian equation, and read off the direction of travel and any excluded points, with worked projectile, circle, parabola and hyperbola examples.
21 min answer →

Where can a function change sign, and how do we use that to solve inequations and to begin a sketch?

Examine the sign of a function by building a table of test values that dodge around its zeroes and discontinuities, and use the resulting sign pattern to solve inequations and to start a curve sketch
The Year 11 Extension 1 dot point on the sign of a function. Why a function can only change sign at a zero or a discontinuity, how to build a table of signs with test values, the repeated-factor exception, rational functions with vertical asymptotes, and using the sign pattern both to solve inequations and to begin a sketch.
17 min answer →