hsc exam questions (real functions)
DESCRIPTION
A collection of questions relating to the topic Real Functions from the 1995-2014 HSC Mathematics Advanced (2U) examinations.TRANSCRIPT
HSC Exam Questions (Real Functions) 1995 HSC Q4(b) 4
(i) Draw the graphs of 𝑦 = |𝑥| and 𝑦 = 𝑥 + 4 on the same set of axes.
(ii) Find the coordinates of the point of intersection of these two graphs.
1997 HSC Q4(b) 6
(i) Sketch the graph of 𝑦 = 𝑥! − 6 and label all intercepts with the axes.
(ii) On the same set of axes, carefully sketch the graph of 𝑦 = |𝑥|.
(iii) Find the 𝑥 coordinates of the two points where the graphs intersect.
(iv) Hence solve the inequality 𝑥! − 6 ≤ |𝑥|.
2000 HSC Q1(g)
Sketch the line 𝑦 = 2𝑥 + 3 in the Cartesian plane. 2
2001 HSC Q5(a)
State the domain and range of the function 𝑦 = 2 25 − 𝑥!. 3
2002 HSC Q6(a)
Sketch the graph of 𝑦 = 4 − 𝑥!, and state the range. 2
2003 HSC Q3(c)
Shade the region in the Cartesian plane for which the inequalities 𝑦 < 𝑥 − 2, 𝑦 ≥ 0 and 𝑥 ≥ 6
hold simultaneously. 2
2008 HSC Q8(a)
Let 𝑓 𝑥 = 𝑥! − 8𝑥!.
(i) Find the coordinates of the points where the graph of 𝑦 = 𝑓(𝑥) crosses the 𝑥 and 𝑦 axes.
1
(ii) Show that 𝑓(𝑥) is an even function. 1
(iii) Sketch the graph of 𝑦 = 𝑓(𝑥). 1
2009 HSC Q1(a)
Sketch the graph of 𝑦 − 2𝑥 = 3, showing the intercepts on both axes. 2
2009 HSC Q3(c)
Shade the region in the plane defined by 𝑦 ≥ 0 and 𝑦 ≤ 4 − 𝑥!. 2
2010 HSC Q1(c)
Write down the equation of the circle with centre (−1, 2) and radius 5. 1
2010 HSC Q1(g)
Let 𝑓 𝑥 = 𝑥 − 8. What is the domain of 𝑓(𝑥)? 1
2010 HSC Q4(d)
Let 𝑓 𝑥 = 1 + 𝑒! .
Show that 𝑓 𝑥 ×𝑓 −𝑥 = 𝑓 𝑥 + 𝑓(−𝑥). 2
2011 HSC Q4(e)
The diagram shows the graphs 𝑦 = 𝑥 − 2 and 𝑦 = 4 − 𝑥!. 2
Write down inequalities that together describe the shaded region.
2013 HSC Q3
Which inequality defines the domain of the function 𝑓 𝑥 = !!!!
?
(A) 𝑥 > −3
(B) 𝑥 ≥ −3
(C) 𝑥 < −3
(D) 𝑥 ≤ −3
2013 HSC Q11(g)
Sketch the region defined by 𝑥 − 2 ! + 𝑦 − 3 ! ≥ 4. 3
2013 HSC Q15(c)
(i) Sketch the graph 𝑦 = |2𝑥 − 3|. 1
(ii) Using the graph from part (i), or otherwise, find all values of 𝑚 for which the equation
2𝑥 − 3 = 𝑚𝑥 + 1 has exactly one solution. 2
2014 HSC Q2
Which graph best represents 𝑦 = 𝑥 − 1 !?
(A)
(B)
(C)
(D)