THETA FUNCTIONS
2016 FAMAT State Convention
The abbreviation NOTA means
"None of the Above Answers."
2
1
and g ( x) 
x
x 1
  1 
find f  g    .
  2 
2
A.
B. 2
3
9
C. 3
D.
2
E. NOTA
ax 2  4
has three
x2  b
asymptotes, two of which have equations
x  3 and y  2 . If a and b are integers,
then give the value of f (1) .
3
A. 2
B. 
4
3
C.
D. 4
5
E. NOTA
5. The graph of f ( x) 
1. For f ( x ) 
2. For f ( x)  x 2 , the equation f ( x)  100
has solutions x  a and x  b . The
equation f ( x)  25 has solutions x  c
and x  d . Give the least possible value
of a  d .
A. 5 3 B. 5
C. 0
D. 5
E. NOTA
3. Given 𝑓(𝑥) = 2𝑥+1 ∙ 4𝑥+2 and 𝑔(𝑥) =
f ( x)
43−𝑥 ∙ 8𝑥 , which is equal to h( x) 
?
g ( x)
A. h( x)  22 x 1 B. h( x)  22 x1
C. h( x)  4 x 1
D. h( x)  4 x 1
E. NOTA
4. For f (2 x  5)  x 2  2 x , find f (7) .
A. 35
B. 14
C. 3
D. 1
E. NOTA
6. C (n)  (n  2)10 . If the expansion of
C (n) is written in standard form with first
term n10 , find the coefficient of the 7th
term of C (n) .
A. 13440 B. 1536
C. 210
D. 120
E. NOTA
7. The graphs of f ( x)  x  3  1 and
g ( x)  3  x  5 intersect at the point
with x-coordinate k . Find g ( k ) .
A. 1
B. 2
C. 3
D. 6
E. NOTA
8. The function f ( x)  x3  3x 2  10 x  24
has real roots a , b, and c for a  b  c .
Give the value of b .
A. 2
B. 3
C. 4
D. 6
E. NOTA
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THETA FUNCTIONS
2016 FAMAT State Convention
9.
y






x











10. What is the domain of y  f ( g ( x)) for
4
1
f ( x) 
and g ( x) 
?
x 1
x2
A. {𝑥 ∈ ℝ|𝑥 ≠ −2 𝑎𝑛𝑑 𝑥 ≠ 1}
B. {𝑥 ∈ ℝ|𝑥 ≠ 2 𝑎𝑛𝑑 𝑥 ≠ 1}
C. {𝑥 ∈ ℝ|𝑥 ≠ −2 𝑎𝑛𝑑 𝑥 ≠ −1}
D. {𝑥 ∈ ℝ|𝑥 ≠ 2 𝑎𝑛𝑑 𝑥 ≠ −1}
E. NOTA


The graph of y  f ( x) is shown.
Which graph below is that of y  f  x  ?
  
f f 2781 ?
y

A.
11. For f ( x)  3 x , which is equal to
A. 39
B. 9 27
C. 27 9 D. 93
E. NOTA






x








12. The function f is cubic, with zeros
2 , i and  i . If f ( x)  x3  bx 2  cx  d
then find f (b ) .
A. 2 B. 4
C. 8 D. 20
E. NOTA




y
B.







x















13. The function d ( x ) is equal to the
number of diagonals in a convex polygon
with x sides (that is, a convex x-gon).
The value of d (3) is 0 because a triangle
has no diagonals. Find d (16)  d (13) .
A. 42
B. 39
C. 32
D. 3
E. NOTA

y



C.




x
















y





D.


x
















E. NOTA
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THETA FUNCTIONS
2016 FAMAT State Convention
Use the table below for questions #14-16.
x
f ( x)
g ( x)
1
2
3
5
1
-3
3
1.5
4
5
-10
-11
1
2
1
1
1
x  x 2  x 3  ... , the
2
4
8
3
equation f ( n)  has solution n 
4
2
1
A. 
B. 
3
6
3
C.
D. 6
2
E. NOTA
17. For f ( x)  1 
2
3
f and g are functions with domains over
the real numbers.
18. f is the function whose graph is the
graph of x 2  y 2  9 for y  0 . g is the
function whose graph is the graph of
( x  1) 2 y 2

 1 for y  0 . Find the value
25
9
of f ( 5)  g (3) .
37
19
A.
B.
5
5
1
17
C.
D. 
5
5
E. NOTA
14. Use the table above to find
g ( f (2))  f ( g (4)) .
5
A.
B. 6
3
19
C.
D. 8
3
E. NOTA
15. Given that f has an inverse y  f 1 ( x)
use the values in the table above
question #14 to find f 1 (1)  f 1 (5) .
6
A. 0
B.
55
6
C.
D. 3
5
E. NOTA
16. If f ( g (k ))  7  f (4) for exactly one
value of k , then use the table to find
the value of k .
A. 3
B. 1
C. 2
D. 3
E. NOTA
19. Given: f ( x, y )  3x  4 y and
g ( x, y )  2 x  y and f (1, k )  g (k ,33) .
Give the value of k .
A. 5
B. 4
C. 4 D. 5
E. NOTA
20. A rhombus has diagonals of lengths
6 and n. The function p ( n) gives the
perimeter of the rhombus. Which is an
expression for p ( n) , over domain n  0 .
A. 2 36  n2
B. 12  n
n2
4
E. NOTA
D. 3 
C.
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9
n
2
THETA FUNCTIONS
2016 FAMAT State Convention
21. For f ( x)  ( x  3)2  p , for a constant
p , give the greatest possible integer
value of p so that f has two distinct
real roots.
A. 3
B. 2
C. 1
D. 0
E. NOTA
22. The segment AB has endpoints (0,3)
and (4, 7). AB has perpendicular
bisector line CD . The function d ( k ) gives
25. f ( x)  6 x  6 x  6 x  ... and
g ( x)  6 x  6 x  6 x  ... .
Give the value of f (5)  g (1) .
A. 30 B. 10 6
C. 8
D. 6
E. NOTA
26. f ( x)  log 2 ( x) . Find the value of x
 1
so that f    f  8 x   4 .
 16 
A. 32
B. 8
1
1
C.
D.
8
32
E. NOTA
the distance from the point on CD with
x  coordinate k to the segment AB .
That is, d (5) gives the distance to AB
from the point which has x  coordinate 5
and is on line CD . Find d (3) .
A. 4 2
B. 3 6
C. 4
D. 2
E. NOTA
23. Which is an expression for g ( x ) if
1

g  x  1  x  1 ?
2

A. 2x 1 B. 2x  2
1
1
1
C. x  1 D. x 
2
2
2
E. NOTA
24.
f ( x)  x  5 and g ( x)  4  f ( x) . The
equation g ( x)  1 has four real solutions
a, b, c, and d for a  b  c  d . Give the
value of 𝑏 ∙ 𝑑.
A. 16 B. 20
C. 40 D. 80
E. NOTA
27. Consider the triangle QRS with side
lengths 4,5, x . T ( x ) =1 if QRS is an
acute triangle. T ( x)  2 if QRS is a right
triangle. T ( x)  3 if QRS an obtuse
triangle. T ( x)  0 if QRS is not a
triangle. For which value of x is T ( x)  3 ?
A. 1.2
B. 10
C. 2 11
D. 8.9
E. NOTA
28. The function L(r ) gives the x-intercept
of the graph of the linear equation
2x  y  r . M (r ) gives the y-intercept of
the graph of the linear equation
2 x  3 y  r  6 . Give the value of r
so that L(r )  M (r ) .
12
A. 
B. 4
5
C. 6
D. 12
E. NOTA
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THETA FUNCTIONS
2016 FAMAT State Convention
29. A lottery ticket has 3 distinct integers
from 1 to 10, inclusive. Ralph will buy a
ticket. The weekly winning ticket has 3
distinct and randomly chosen integers
from 1 to 10, inclusive. The function
W ( k ) gives the amount in dollars that
Ralph will win if he matches exactly k
numbers on his ticket to the official
winning ticket. W(1)=20, W(2)=200, and
W(3)=2000. Order is not important. So if
Ralph's ticket shows 1, 5 ,6 and the
winning ticket shows 1, 6, 8 then k=2.
Find the probability that W ( x)  200
1
1
A.
B.
70
40
7
7
C.
D.
120
40
E. NOTA
30. The solution to the system
a x  b1 y  c1
of linear function  1
is
a2 x  b2 y  c2
( x, y ) . When Cramer's Rule is used
to find the solution to such a system in
terms of x and y, the value of x is given
31 7
by x 
2
1
. Find ( x, y ) .
3 7
2
1
 1 
A. 1, 
 17 
B.
C. (1, 7)
2 

D.  1, 
 17 
 1, 4
E. NOTA
Page 5