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MAT 111 Tutorial 2

The document contains tutorial questions for a mathematics course covering topics such as functions, relations, composition of functions, equations of lines, and properties of functions. It includes exercises on defining functions, determining if relations are functions, finding compositions, and solving for constants in function definitions. Additionally, it addresses graphing functions, analyzing discriminants, and applying mathematical concepts to real-world scenarios.

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28 views3 pages

MAT 111 Tutorial 2

The document contains tutorial questions for a mathematics course covering topics such as functions, relations, composition of functions, equations of lines, and properties of functions. It includes exercises on defining functions, determining if relations are functions, finding compositions, and solving for constants in function definitions. Additionally, it addresses graphing functions, analyzing discriminants, and applying mathematical concepts to real-world scenarios.

Uploaded by

protodesire1
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MAT 111 Tutorial Questions MUST

1. (a) Define a function.


(b) Let X = {1, 2, 3, 4}. Determine whether each relation on X is a function from X
into X.
i. f = {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)},
ii. g = {(3, 1), (4, 2), (1, 1)},
iii. h = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}.
(c) Let A = {a, b, c}, B = {x, y, z}, C = {r, s, t}. Let f: A→B and g: B→C be defined
by: f = {(a, y)(b, x), (c, y)} and g = {(x, s), (y, t), (z, r)}.
Find the composition function g◦f: A→C.

2. (a) Let f : R → R and g: R → R be defined by f (x) = 2x + 1 and g(x) = x2 − 2. Find


the formula for the composition function g(f (x)) .
(b) Let the functions f: A → B, g: B → C, h: C → D be defined by the fig. below.
Determine if each function is:
i. onto,
ii. one-to-one,
iii. invertible.

3. Find the equation of the line


(a) through(-2,3) with gradient − 21 ,
(b) through the points (-3,2) and (-1,-5),
(c) through (-1,-1) perpendicular to the line 3x − 2y = 1.

4. The line y = 2x + 3 intersects the y-axis at A. The points B and C on this line are
such that AB=BC. The line through B perpendicular to AC passes through the point
D(-1,6). Find
(a) the equation of BD,
(b) the coordinates of B,
(c) the coordinates of C.

5. The line y = x + 2 meets the curve y 2 = 4(2x + 1) at A and B. Find the coordinates of
the midpoint of AB.

Week: 23-27 June 2014


6. The functions f and g are defined for x ∈ R by

f (x) = 3x + a

g(x) = b − 2x
where a and b are constants. Given that f (f (2)) = 10 and g −1 (2) = 3, find
(a) the values of a and b,
(b) an expression for f (g(x)).

7. Express in terms of functions f (x) = x + 3 and g(x) = x2 ,


(a) x2 + 3,
(b) x2 + 6x + 9,
(c) x + 6,
(d) x2 + 6x + 12,
(e) x2 − 6x + 9.

8. The function f is defined by

f : x → 2x2 − 8x + 10 for 0 ≤ x ≤ 2,

(a) Express f (x) in the form a(x + b)2 + c, where a, b and c are constants.
(b) State the range of f (x).
(c) State the domain of f −1 (x).
(d) Find an expression for f −1 (x).
(e) Sketch f (x).

9. Use the graph of f (x) = x2 to sketch the following graphs


(a) g(x) = (x + 1)2 ,
(b) h(x) = (x − 3)2 ,
(c) j(x) = x2 − 6x + 6.

10. (a) If f (x) = ax2 + bx + c explain what is meant by the discriminant of f (x).
(b) Use the discriminant to predict the number of horizontal intercepts for the function
f (x) = x2 + 2x + 2.
(c) For what values of m does the equation 4x2 + 8x − 8 = m(4x − 3) have no real
roots?
(d) Find the maximum or minimum values of the following functions and the values of
x where this occurs:
i. x2 − 6x − 1,
ii. 3 − x − 2x2 .
11. The performance record of a student learning to read is given by N (t) = 100 (1 − e−0.2t ),
where N is the reading speed (words per minute), and t is the number of weeks of
instruction. Find
(a) the reading speed after 40 weeks,
(b) time for speed to reach 50 words per minute,
(c) Sketch the graph the function for 0 ≤ t ≤ 50.

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