Functions LAST Paper1 [133 marks]
1a. [1 mark]
The following table shows values of 𝑓(𝑥) and 𝑔(𝑥) for different values of 𝑥.
Both 𝑓 and 𝑔 are one-to-one functions.
Find 𝑔(0).
1b. [2 marks]
Find (𝑓 ∘ 𝑔)(0).
1c. [2 marks]
Find the value of 𝑥 such that 𝑓(𝑥) = 0.
2a. [1 mark]
!"#$
A function 𝑓 is defined by 𝑓(𝑥) = "%$
, where 𝑥 ∈ ℝ, 𝑥 ≠ −1.
The graph of 𝑦 = 𝑓(𝑥) has a vertical asymptote and a horizontal asymptote.
Write down the equation of the vertical asymptote.
2b. [1 mark]
Write down the equation of the horizontal asymptote.
2c. [3 marks]
On the set of axes below, sketch the graph of 𝑦 = 𝑓(𝑥).
On your sketch, clearly indicate the asymptotes and the position of any points
of intersection with the axes.
2d. [1 mark]
!"#$
Hence, solve the inequality 0 < "%$
< 2.
3a. [1 mark]
Write down the equation of the vertical asymptote.
3b. [1 mark]
Write down the equation of the horizontal asymptote.
3c. [3 marks]
On the set of axes below, sketch the graph of 𝑦 = 𝑓(𝑥).
On your sketch, clearly indicate the asymptotes and the position of any points
of intersection with the axes.
3d. [1 mark]
!"#$
Hence, solve the inequality 0 < "%$
< 2.
3e. [2 marks]
!|" |#$
Solve the inequality 0 < |"|%$
< 2.
4a. [2 marks]
Consider the function 𝑓(𝑥) = −2(𝑥 − 1)(𝑥 + 3), for 𝑥 ∈ ℝ. The following diagram shows
part of the graph of 𝑓.
For the graph of 𝑓
find the 𝑥-coordinates of the 𝑥-intercepts.
4b. [3 marks]
find the coordinates of the vertex.
4c. [2 marks]
The function 𝑓 can be written in the form 𝑓(𝑥) = −2(𝑥 − ℎ)! + 𝑘.
Write down the value of ℎ and the value of 𝑘.
5a. [1 mark]
!"%'
The function 𝑓 is defined by 𝑓(𝑥) = (#"
, where 𝑥 ∈ ℝ, 𝑥 ≠ 3.
Write down the equation of
the vertical asymptote of the graph of 𝑓.
5b. [1 mark]
the horizontal asymptote of the graph of 𝑓.
5c. [1 mark]
Find the coordinates where the graph of 𝑓 crosses
the 𝑥-axis.
5d. [1 mark]
the 𝑦-axis.
5e. [1 mark]
Sketch the graph of 𝑓 on the axes below.
6a. [1 mark]
the vertical asymptote of the graph of 𝑓.
6b. [1 mark]
the horizontal asymptote of the graph of 𝑓.
6c. [1 mark]
the 𝑥-axis.
6d. [1 mark]
the 𝑦-axis.
6e. [1 mark]
Sketch the graph of 𝑓 on the axes below.
6f. [4 marks]
)"%'
The function 𝑔 is defined by 𝑔(𝑥) = (#"
, where 𝑥 ∈ ℝ, 𝑥 ≠ 3 and 𝑎 ∈ ℝ.
Given that 𝑔(𝑥) = 𝑔#$ (𝑥), determine the value of 𝑎.
7a. [5 marks]
The equation 3𝑝𝑥 ! + 2𝑝𝑥 + 1 = 𝑝 has two real, distinct roots.
Find the possible values for 𝑝.
7b. [2 marks]
Consider the case when 𝑝 = 4. The roots of the equation can be expressed in the form 𝑥 =
)±√$(
,
, where 𝑎 ∈ ℤ. Find the value of 𝑎.
8a. [1 mark]
The graph of 𝑦 = 𝑓(𝑥) for −4 ≤ 𝑥 ≤ 6 is shown in the following diagram.
Write down the value of 𝑓(2).
8b. [1 mark]
Write down the value of (𝑓 ∘ 𝑓)(2).
8c. [3 marks]
$
Let 𝑔(𝑥) = ! 𝑓(𝑥) + 1 for −4 ≤ 𝑥 ≤ 6. On the axes above, sketch the graph of 𝑔.
9a. [6 marks]
Let 𝑓(𝑥) = 𝑚𝑥 ! − 2𝑚𝑥, where 𝑥 ∈ ℝ and 𝑚 ∈ ℝ. The line 𝑦 = 𝑚𝑥 − 9 meets the graph
of 𝑓 at exactly one point.
Show that 𝑚 = 4.
9b. [2 marks]
The function 𝑓 can be expressed in the form 𝑓(𝑥) = 4(𝑥 − 𝑝)(𝑥 − 𝑞), where 𝑝, 𝑞 ∈ ℝ.
Find the value of 𝑝 and the value of 𝑞.
9c. [3 marks]
The function 𝑓 can also be expressed in the form 𝑓(𝑥) = 4(𝑥 − ℎ)! + 𝑘, where ℎ, 𝑘 ∈ ℝ.
Find the value of ℎ and the value of 𝑘.
9d. [3 marks]
Hence find the values of 𝑥 where the graph of 𝑓 is both negative and increasing.
10. [5 marks]
The cubic equation 𝑥 ( − 𝑘𝑥 ! + 3𝑘 = 0 where 𝑘 > 0 has roots 𝛼, 𝛽 and 𝛼 + 𝛽.
-!
Given that 𝛼𝛽 = − '
, find the value of 𝑘.
11. [6 marks]
Let 𝑓(𝑥) = −𝑥 ! + 4𝑥 + 5 and 𝑔(𝑥) = −𝑓(𝑥) + 𝑘.
Find the values of 𝑘 so that 𝑔(𝑥) = 0 has no real roots.
12a. [3 marks]
Let 𝑔(𝑥) = 𝑥 ! + 𝑏𝑥 + 11. The point (−1, 8) lies on the graph of 𝑔.
Find the value of 𝑏.
12b. [4 marks]
The graph of 𝑓(𝑥) = 𝑥 ! is transformed to obtain the graph of 𝑔.
Describe this transformation.
13. [5 marks]
Consider the function 𝑓(𝑥) = 𝑥 ' − 6𝑥 ! − 2𝑥 + 4, 𝑥 ∈ ℝ.
The graph of 𝑓 is translated two units to the left to form the function 𝑔(𝑥).
Express 𝑔(𝑥) in the form 𝑎𝑥 ' + 𝑏𝑥 ( + 𝑐𝑥 ! + 𝑑𝑥 + 𝑒 where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 ∈ ℤ.
14. [5 marks]
"#'
Sketch the graph of 𝑦 = !"#., stating the equations of any asymptotes and the coordinates
of any points of intersection with the axes.
15a. [2 marks]
"%(
The functions 𝑓 and 𝑔 are defined such that 𝑓(𝑥) = '
and 𝑔(𝑥) = 8𝑥 + 5.
Show that (𝑔 ∘ 𝑓)(𝑥) = 2𝑥 + 11.
15b. [3 marks]
Given that (𝑔 ∘ 𝑓)#$ (𝑎) = 4, find the value of 𝑎.
16. [5 marks]
The following diagram shows the graph of 𝑦 = 𝑓(𝑥). The graph has a horizontal
asymptote at 𝑦 = −1. The graph crosses the 𝑥-axis at 𝑥 = −1 and 𝑥 = 1, and the 𝑦-axis
at 𝑦 = 2.
On the following set of axes, sketch the graph of 𝑦 = [𝑓(𝑥)]! + 1, clearly showing
any asymptotes with their equations and the coordinates of any local maxima or minima.
17. [6 marks]
The functions 𝑓 and 𝑔 are defined for 𝑥 ∈ ℝ by 𝑓(𝑥) = 𝑥 − 2 and 𝑔(𝑥) = 𝑎𝑥 + 𝑏,
where 𝑎, 𝑏 ∈ ℝ.
Given that (𝑓 ∘ 𝑔)(2) = −3 and (𝑔 ∘ 𝑓)(1) = 5, find the value of 𝑎 and the value of 𝑏.
18a. [3 marks]
The following diagram shows the graph of 𝑦 = −1 − √𝑥 + 3 for 𝑥 ≥ −3.
Describe a sequence of transformations that transforms the graph of 𝑦 = √𝑥 for 𝑥 ≥
0 to the graph of 𝑦 = −1 − √𝑥 + 3 for 𝑥 ≥ −3.
18b. [1 mark]
A function 𝑓 is defined by 𝑓(𝑥) = −1 − √𝑥 + 3 for 𝑥 ≥ −3.
State the range of 𝑓.
18c. [5 marks]
Find an expression for 𝑓 #$ (𝑥), stating its domain.
18d. [5 marks]
Find the coordinates of the point(s) where the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑓 #$ (𝑥)
intersect.
19a. [4 marks]
!" ! #."#$!
Let 𝑓(𝑥) = "%!
, 𝑥 ∈ ℝ, 𝑥 ≠ −2.
Find all the intercepts of the graph of 𝑓(𝑥) with both the 𝑥 and 𝑦 axes.
19b. [1 mark]
Write down the equation of the vertical asymptote.
19c. [4 marks]
As 𝑥 → ±∞ the graph of 𝑓(𝑥) approaches an oblique straight line asymptote.
Divide 2𝑥 ! − 5𝑥 − 12 by 𝑥 + 2 to find the equation of this asymptote.
