IB HL Math Homework #4: Functions and Equations Assigned: 9/20/07, Thursday

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IB HL Math Homework #4: Functions and Equations
Assigned: 9/20/07, Thursday
Due: 9/25/07, Tuesday
1) Find all of the values of x that satisfy the inequality
2x
 1.
| x 1|
2) The polynomial f(x) = x3 + 3x2 + ax + b leaves the same remainder when divided by (x
– 2) as when divided by (x + 1). Find the value of a.
3) The functions f and g are defined by f : x  e x , g : x  x  2. Calculate
(a) f 1 (3)  g 1 (3)
(b) ( f  g ) 1 (3).
4) A sum of $5000 is invested at a compound interest rate of 6.3% per annum.
(a) Write down an expression for the value of the investment after n full years.
(b) What will be the value of the investment at the end of five years?
(c) The value of the investment will exceed $10000 after n full years.
(i) Write an inequality to represent this information.
(ii) Calculate the minimum value of n.
5) Let f ( x) 
f ( x)  g ( x) .
x4
x2
, x  1 and g ( x) 
, x  4 . Find the set of values of x such that
x 1
x4
6) Given that (x – 1) and (x – 2) are factors of 6x4 + ax3 – 17x2 + bx – 4, find a and b,
and any remaining factors.
7) Consider the functions f ( x)  e 2 x and g ( x)  sin
x
2
.
(a) Find the period of the function f  g.
(b) Find the intervals for which ( f  g ( x))  4.
8) The function f is defined on the domain [-1,0] by f : x 
(a) Write done the range of f.
1
.
1 x2
(b) Find an expression for f-1(x).
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