Further work with functions
What this note covers
- Reciprocal graphs and reciprocal trigonometric functions
- Square roots, y² = f(x), sums and products of functions
- Inverse functions
- Parametric form and inequalities
4 sections · 7 key terms & formulas · 6 common mistakes
Extension 1 asks you to build new graphs out of a graph you already know. The first and most important transformation is the reciprocal: given y = f(x), sketch y = 1/f(x). Rather than plotting points, you read features straight off the original curve.
- Zeros become vertical asymptotes. Wherever f(x) = 0, the reciprocal is undefined and its magnitude grows without bound, so draw a dashed vertical line there.
- Large values become small values. If f(x) → ±∞, then 1/f(x) → 0. In particular, if f(x) grows without bound as x → ±∞, the reciprocal has the horizontal asymptote y = 0. If f(x) → k (k ≠ 0), then 1/f(x) → 1/k.
- Sign is preserved. 1/f(x) is positive exactly where f(x) is positive, so the reciprocal stays on the same side of the x-axis.
- The values 1 and −1 are fixed. Where f(x) = ±1, the two graphs meet. These are useful anchor points.
- Turning points stay at the same x-value. A local maximum of f with a non-zero value becomes a local minimum of 1/f, and vice versa, because a larger positive number has a smaller reciprocal.
- Increasing becomes decreasing on any interval where f does not change sign.
Start with f(x) = x² − 4, a concave-up parabola with zeros at x = ±2 and minimum value −4 at x = 0.
Step 1: zeros x = ±2 become vertical asymptotes x = −2 and x = 2.
Step 2: as x → ±∞, f(x) → ∞, so 1/f(x) → 0⁺; the horizontal asymptote is y = 0, approached from above.
Step 3: the minimum (0, −4) of f becomes a local maximum (0, −1/4) of the reciprocal.
Step 4: signs. For |x| > 2, f > 0, so the outer branches lie above the x-axis. For −2 < x < 2, f < 0, so the middle branch is below the axis, an upside-down ‘cap’ peaking at (0, −1/4) and plunging to −∞ at each asymptote.
Domain: x ≠ ±2. Range: y ≤ −1/4 or y > 0.
Reciprocal trigonometric functions. The three reciprocal ratios are cosec x = 1/sin x, sec x = 1/cos x and cot x = 1/tan x = cos x / sin x. Apply the rules above to the familiar sine, cosine and tangent graphs.
- y = cosec x has vertical asymptotes where sin x = 0, that is x = nπ (or 180°n). It touches y = 1 at x = π/2 + 2nπ and y = −1 at x = 3π/2 + 2nπ. Range: y ≤ −1 or y ≥ 1. Period 2π.
- y = sec x has asymptotes where cos x = 0, that is x = π/2 + nπ. It has local minima of 1 at x = 2nπ and local maxima of −1 at x = π + 2nπ. Range: y ≤ −1 or y ≥ 1. Period 2π. It is an even function, like cosine.
- y = cot x has asymptotes at x = nπ, zeros at x = π/2 + nπ, is decreasing on each branch and has period π. Its range is all real numbers.
Domain: cos x ≠ 0, so x ≠ π/2, 3π/2 within [0, 2π]. Since sec x ≤ −1 or sec x ≥ 1, the range of 2 sec x is y ≤ −2 or y ≥ 2.
Solve: sec x = 2 means cos x = 1/2, so x = π/3 or x = 5π/3. Both lie in the domain.
When working in degrees the same shapes apply, with asymptotes at multiples of 90° or 180°. Always label asymptotes with dashed lines and mark the points where the reciprocal meets the original curve at y = ±1.
Common questions
What does the HSC Year 11 Mathematics Extension 1 note "Further work with functions" cover?
This note has 4 sections on Further work with functions. It includes 7 key terms and formulas, plus 6 common mistakes. It covers 11, Functions: Further work with functions (ME1-11-01). The full section list is shown under “What this note covers”.
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