A. Armstrong
MA 141- Fall 2012 - Pre-Calculus Review
1. Simplify the rational expression:
y
x
1
y
−
−
x
y
1
x
.
2. State whether each equation is true or false. If false, explain why or correct the mistake.
(a) (p + q)2 = p2 + q 2
√
(b) a2 + b2 = a + b
√
√ √
(c) ab = a b
1
1 1
(d)
= −
x−y
x y
(e)
1/x
1
=
a/x − b/x
a−b
(f)
1 + TC
=1+T
C
3. The graphs of f and g are given.
(a)
(b)
(c)
(d)
State the values of f (−4) and g(3).
For what values of x is f (x) = g(x)?
On what interval is f increasing?
State the domain and range of both f and g.
4. If f (x) = 3x2 − x + 2, find f (2), f (a), f (a2 ), [f (a)]2 , and f (a + h).
5. Evaluate the difference quotient for each function. Simplify your answer.
(a) f (x) = x3 ;
(b) f (x) =
1
;
x
f (a + h) − f (a)
h
f (x) − f (a)
x−a
6. Find the domain
and sketch the graph of the following functions.
√
(a) g(x) = 5 − x
(b) H(t) =
4 − t2
2−t
1
7. Sketch the graph of the following function:

 x + 9 if x < −3;
−2x if | x |≤ 3;
f (x) =

−6
if x > 3
8. At the surface of the ocean, the water pressure is the same as the air pressure above the
water, 15 lb/in2 . Below the surface, the water pressure increases by 4.34 lb/in2 for every
10 ft of descent.
(a) Express the water pressure as a function of the depth below the ocean surface.
(b) At what depth is the pressure 100 lb/in2 ?
9. Graph the functions by hand, not by plotting points, but by starting with the graph of a
standard function and then applying the appropriate transformations.
(a) y = 1 − x2
(b) y = 4 sin 3x
10. Given functions, f and g, find the following:
f (x) = 1 − 3x;
g(x) = cos x
(a) f ◦ g
(b) g ◦ f
(c) f ◦ f
11. Use the laws of exponents to simplify the following:
x2n x3n−1
xn+2
12. The half-life of bismuth-210 is 5 days. Suppose a sample of bismuth-210 has a mass of
200 mg.
(a) What is the amount remaining after 15 days?
(b) Estimate the amount remaining after 3 weeks.
(c) How long will it take for the mass to be reduced to 1 mg?