Sample Exam
1
1. (20 points) Let f (x) = 4x 1, x R g(x) = 3 2x , x R, x = 3 ,
2
2
h(x) = 3x 6x + 4, x R.
(a) Find the value of f (2), g(8), h(1),
(b) Compute g(f (x), h(f (x).
(c) Sketch f (x) and h(x). State their domain and range.
2. (20 points) Solve t
Functions III1
1
The Exponential Function
Denition 1 A function f (x) has exponential growth if
f (x + 1)
= k is a conf (x)
stant k > 1, not de- f (x) pending on x.
This means that for any x, if you increase x by 1, you multiply the y-value f (x)
by k to
Exercise 3: State the domain of each function given its range:
(a) f (x) = x2 + 2x 8,
f (x) R. This is a trivial question. Since there are
not restrictions in f (x), x can take any value in the real numbers. So the
domain is D = R
1 g(x) < 6. First we obs
Integration
1 Remark. There are two topics with similar names:
Reverse of differentiation
Indenite integral
f (x) dx = most general antiderivative for f (x)
Denite integral
This is related to summation (it is a limit of sums of a certain kind). The inte
1
MA1MO1, Solution Sheet 2.
4
Exercise 1: Let f (x) = 3x 5, x R g(x) = 6 x , x R, x = 6 (Explain to
the students why we cant have x = 6 in the domain), h(x) = x2 + 4x 1, x R.
Find the value of
(a) f (3), g(4), h(1)
f (3) = 3 3 5 = 9 5 = 4
g(4) =
4
4
= =2
MA1MO1, Solution Sheet 3.1
Exercise 1: Find the equation of the following straight lines. What is the slope?
and the x and y-intercepts? Sketch them.
(a) The straight line that passes through (1, 2) and (3, 6).
The slope is given by:
m=
62
4
= =1
3 (1)
4
MA1MO1, Solution Sheet 5. (Homework)1
Exercise 1: Compute cosine, sine, tangent, cotangent, secant and cosecant of
the following angles:
(b) x = 5 . First notice that 5 = 0.8333333. This means that we are in
6
6
the second quadrant. The angle in the rst q