MAT1510
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DISCLAIMER & TERMS OF USE
Educational Aid: These study notes are intended to be used as educational resources and should not be seen as a
replacement for individual research, critical analysis, or professional consultation. Students are encouraged to perform
their own research and seek advice from their instructors or academic advisors for specific assignment guidelines.
Personal Responsibility: While every effort has been made to ensure the accuracy and reliability of the information in
these study notes, the seller does not guarantee the completeness or correctness of all content. The buyer is
responsible for verifying the accuracy of the information and exercising their own judgment when applying it to their
assignments.
Academic Integrity: It is essential for students to maintain academic integrity and follow their institution's policies
regarding plagiarism, citation, and referencing. These study notes should be used as learning tools and sources of
inspiration. Any direct reproduction of the content without proper citation and acknowledgment may be considered
academic misconduct.
Limited Liability: The seller shall not be liable for any direct or indirect damages, losses, or consequences arising from
the use of these notes. This includes, but is not limited to, poor academic performance, penalties, or any other negative
consequences resulting from the application or misuse of the information provided.
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ASSIGNMENT 02
MAT1510 EXAM 2025
Due date: Friday, 31 May 2024
Total Marks: 100
UNIQUE ASSIGNMENT NUMBER: 186115
ONLY FOR YEAR MODULE
This assignment covers chapter 2 of the prescribed book as well as the study guide
DO NOT USE A CALCULATOR.
Question 1: 13 Marks
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give
an example that disproves the statement.
(1.1) If f is a function, then f (s + t) = f (s) + f (t). (3)
(1.2) If f (s) = f (t), then s = t. (2)
(1.3) If f is a function, then f (3x) = 3f (x). (3)
(1.4) A vertical line intersects the graph of a function at most once. (2)
1
(1.5) If f is one-to-one, then f −1 (x) = . (3)
f (x)
Question 2: 9 Marks
The perimeter of a rectangle is 16 meters.
(2.1) If the length of one of the sides of the rectangle is (1 + x) meters, express the area A of the (4)
rectangle in terms of x.
(2.2) Calculate the maximum area of the rectangle. (3)
(2.3) What are the dimensions of the rectangle when its area is a maximum? (2)
Question 3: 6 Marks
Suppose a stone is thrown vertically upwards with a velocity of u meters per second. Then its height is h (in
meters) after t seconds is given by the formula
h = ut − 4.8t 2 .
(3.1) Suppose the stone is thrown upwards with a velocity of 24 meters per second. Sketch the (5)
graph of the function defined by
h = ut − 4.8t 2 .
Label the axes properly, and show the coordinates of the critical points on the graph clearly.
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MAT1510/101/0/2023
(3.2) What is the maximum height that the stone reaches? (1)
Question 4: 6 Marks
Sketch the graph of the function g(x) which is piecewise-defined by
(4.1) (4)
2
−x + 2x + 3
if x < 1
g(x) = 4 if x = 1
2
x − 2x + 5 if x > 1
(4.2) Explain why g is called a function. (1)
(4.3) Is g a one-to-one function? Give a reason for your answer. (1)
Question 5: 20 Marks
The sketch shows the graph of the functions f and g. Function f is defined by
y = f (x) = m|x − p| + q
17
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and g is defined by
y = g(x) = ax 2 + bx + c
S is the salient point of the graph of f , and T is the turning point of the graph of g. The two graphs intersect
each other at T and R.
(5.1) Determine the value of m, p and q and then write down the equation of f . (4)
(5.2) Describe the steps of the transformation process that you would apply to the graph of f to (3)
obtain the graph of y = 5|x|.
(5.3) Find the values of a, b and c and then write down the equation of g. (4)
(5.4) Determine the coordinates of T . (2)
(5.5) Use the graph of f and g to solve the inequality (3)
f (x)
<1 for x ∈ (−2, 4).
g(x)
(Do not solve the inequality algebraically.)
(5.6) Suppose the function d describes the vertical distance between the graph of f and g on the
interval [−2, xT ] ( where xT is the x-coordinate of T ).
(a) Complete and simplify the equation (2)
d(x) = ................ for x ∈ [−2, xT ].
(b) What is the maximum vertical distance between the graphs of f and g on the interval (2)
[−2, xT ].
Question 6: 11 Marks
Suppose a function g is defined by
y = g(x) = 7 + 6x − x 2
(6.1) Restrict the domain of g such that the function gr defined by (2)
gr (x) = g (x) for all x ∈ Dgr ,
is one-to-one function, and such that the domain Dgr contains only positive numbers.
(6.2) Determine the equation of the inverse function gr−1 and the set Dg −1 . (3)
r
(6.3) Show that (6)
gr ◦ gr−1
(x) = x for x ∈ Dg −1
r
and
gr−1 ◦ gr (x) = x
for x ∈ Dgr .
18
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EXAM PACK
Recent exam questions and answers
Summarised study notes
Exam tips and guidelines
+27 81 278 3372
DISCLAIMER & TERMS OF USE
Educational Aid: These study notes are intended to be used as educational resources and should not be seen as a
replacement for individual research, critical analysis, or professional consultation. Students are encouraged to perform
their own research and seek advice from their instructors or academic advisors for specific assignment guidelines.
Personal Responsibility: While every effort has been made to ensure the accuracy and reliability of the information in
these study notes, the seller does not guarantee the completeness or correctness of all content. The buyer is
responsible for verifying the accuracy of the information and exercising their own judgment when applying it to their
assignments.
Academic Integrity: It is essential for students to maintain academic integrity and follow their institution's policies
regarding plagiarism, citation, and referencing. These study notes should be used as learning tools and sources of
inspiration. Any direct reproduction of the content without proper citation and acknowledgment may be considered
academic misconduct.
Limited Liability: The seller shall not be liable for any direct or indirect damages, losses, or consequences arising from
the use of these notes. This includes, but is not limited to, poor academic performance, penalties, or any other negative
consequences resulting from the application or misuse of the information provided.
, lOMoARcPSD|53028991
ASSIGNMENT 02
MAT1510 EXAM 2025
Due date: Friday, 31 May 2024
Total Marks: 100
UNIQUE ASSIGNMENT NUMBER: 186115
ONLY FOR YEAR MODULE
This assignment covers chapter 2 of the prescribed book as well as the study guide
DO NOT USE A CALCULATOR.
Question 1: 13 Marks
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give
an example that disproves the statement.
(1.1) If f is a function, then f (s + t) = f (s) + f (t). (3)
(1.2) If f (s) = f (t), then s = t. (2)
(1.3) If f is a function, then f (3x) = 3f (x). (3)
(1.4) A vertical line intersects the graph of a function at most once. (2)
1
(1.5) If f is one-to-one, then f −1 (x) = . (3)
f (x)
Question 2: 9 Marks
The perimeter of a rectangle is 16 meters.
(2.1) If the length of one of the sides of the rectangle is (1 + x) meters, express the area A of the (4)
rectangle in terms of x.
(2.2) Calculate the maximum area of the rectangle. (3)
(2.3) What are the dimensions of the rectangle when its area is a maximum? (2)
Question 3: 6 Marks
Suppose a stone is thrown vertically upwards with a velocity of u meters per second. Then its height is h (in
meters) after t seconds is given by the formula
h = ut − 4.8t 2 .
(3.1) Suppose the stone is thrown upwards with a velocity of 24 meters per second. Sketch the (5)
graph of the function defined by
h = ut − 4.8t 2 .
Label the axes properly, and show the coordinates of the critical points on the graph clearly.
16
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, lOMoARcPSD|53028991
MAT1510/101/0/2023
(3.2) What is the maximum height that the stone reaches? (1)
Question 4: 6 Marks
Sketch the graph of the function g(x) which is piecewise-defined by
(4.1) (4)
2
−x + 2x + 3
if x < 1
g(x) = 4 if x = 1
2
x − 2x + 5 if x > 1
(4.2) Explain why g is called a function. (1)
(4.3) Is g a one-to-one function? Give a reason for your answer. (1)
Question 5: 20 Marks
The sketch shows the graph of the functions f and g. Function f is defined by
y = f (x) = m|x − p| + q
17
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, lOMoARcPSD|53028991
and g is defined by
y = g(x) = ax 2 + bx + c
S is the salient point of the graph of f , and T is the turning point of the graph of g. The two graphs intersect
each other at T and R.
(5.1) Determine the value of m, p and q and then write down the equation of f . (4)
(5.2) Describe the steps of the transformation process that you would apply to the graph of f to (3)
obtain the graph of y = 5|x|.
(5.3) Find the values of a, b and c and then write down the equation of g. (4)
(5.4) Determine the coordinates of T . (2)
(5.5) Use the graph of f and g to solve the inequality (3)
f (x)
<1 for x ∈ (−2, 4).
g(x)
(Do not solve the inequality algebraically.)
(5.6) Suppose the function d describes the vertical distance between the graph of f and g on the
interval [−2, xT ] ( where xT is the x-coordinate of T ).
(a) Complete and simplify the equation (2)
d(x) = ................ for x ∈ [−2, xT ].
(b) What is the maximum vertical distance between the graphs of f and g on the interval (2)
[−2, xT ].
Question 6: 11 Marks
Suppose a function g is defined by
y = g(x) = 7 + 6x − x 2
(6.1) Restrict the domain of g such that the function gr defined by (2)
gr (x) = g (x) for all x ∈ Dgr ,
is one-to-one function, and such that the domain Dgr contains only positive numbers.
(6.2) Determine the equation of the inverse function gr−1 and the set Dg −1 . (3)
r
(6.3) Show that (6)
gr ◦ gr−1
(x) = x for x ∈ Dg −1
r
and
gr−1 ◦ gr (x) = x
for x ∈ Dgr .
18
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