Graded Recitation 1.1

Group Activity Warm-up Exercises 1. If f ( x) =

1 1− x f ( x + h) − f ( x ) and g ( x) = , find (a) f ( g ( x)) and (b) . x +1 x h

f ( − 2) x 2 − x3 2. If f ( x) = and g ( x) = 4 − x , find (a) and (b) g ( g (4 x − x 2 )) . 2 g (0) x+5 3. If f ( x) = x + 2 and g ( x) = x 2 , for what value(s) of x is f ( g ( x )) = g ( f ( x)) ? 4. If f ( x) =

kx + 2 , find k if (−2,1) is an ordered pair of f (x) . x2 + k

5. (COMC 2006) If f ( 2 x + 1) = ( x − 12)( x + 13) , what is f (31) ? Mind Stretching Exercises 6. (AMC 1980) If f (1) = 5 and f ( x + 1) = 2 f ( x) , then what is the value of f (7) ? 7. (USC 1990) For all real numbers x, the function f (x ) satisfies 2 f ( x) + f (1 − x) = x 2 . Find f (5) . 8. (PMO 2008) Let f : R → R be a function such that f ( a + b) = f (a ) + f (b) and

that f ( 2008) = 3012 . What is f ( 2009) ? 9. (PMO 2008) Let f be a function such that f ( 2 − 3 x) = 4 − x . Find the value of 12

∑ f (i) . i =1

10. (UG 2005) Let f (n) = n 3 and define the function g (n) by the formula g (n) = f (n + 1) − f (n) . What is the average of the ten numbers g (0) , g (1) , … , g (9) ? Honors’ Problem (Optional) Let f ( x) = (a − 2) x 2 + (b + 3) x + c and f (0) = f (1) = f (2) = 1 . What is the numerical value of a +b+c ?

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Philippine Mathematical Olympiad 2008 Australian Mathematics Competition 1980 Canadian Open Mathematics Challenge 2006 University of Georgia High School Math Tournament 2005 University of South Carolina High School Math Contest 1990