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Functions

Year 2  »  Pure

4.2 Activity

Instructions

(i)Decide if each of the following functions are many-to-one or one-one.
(ii)If a function is many-to-one, give the greatest possible domain over which the function is one-one. (there may be more than one possible answer).

Notes:

  • For each function, assume that \(x \in \mathbb{R}\).
  • There may be more than one correct answer.

Tip: Consider sketching the graph when you are unsure.


(a)\({\rm{a}}(x) = {x^3}\)
(b)\({\rm{b}}(x) = 3x + 5\)
(c)\({\rm{c}}(x) = {x^2} + 2x - 3,\ \ {\rm{ }}x \ge - 1\)
(d)\({\rm{d}}(x) = \sin 2x,{\rm{ }}\ - \frac{\pi }{2} \le x \le \frac{\pi }{2}\)
(e)\({\rm{e}}(x) = \cos x,\ \ {\rm{ }}0 \le x \le \pi \)
(f)\(\begin{aligned}{\rm{f}}(x) = \frac{1}{{2x - 5}},\ \ {\rm{ }}x \ne 2.5\end{aligned}\)
(g)\({\rm{g}}(x) = - 2{x^2} + 5x - 2,\,\,{\rm{ }}x \ge 0.5\)
(h)\({\rm{h}}(x) = \sqrt {x - 1} ,\,\,x \ge 1\)
(i)\({\rm{i}}(x) = {x^3} - 4x\)
(j*)\(\begin{aligned}{\rm{j}}(x) = \frac{1}{{\ln x}}\ \ \end{aligned}\) (*extension)

Answers

(a)
One-to-one
(b)
One-to-one
(c)
One-to-one
(d)
Many-to-one,   \(-\frac{\pi }{4} \le x \le \frac{\pi }{4}\)
(e)
One-to-one
(f)
One-to-one
(g)
Many-to-one,   \(x \ge \frac{5}{4}\) (alternate answer: \(x \le \frac{5}{4}\))
(h)
One-to-one
(i)
Many-to-one,   \(x \ge \frac{2}{{\sqrt 3 }}\)
(j)
One-to-one