If f(x) = {(2, 5), (6, 9), (11, 2), (x, 4)} is a function, what is the smallest value x may NOT have?
The answer is 2. If f(x) is a function, then each member of the domain is mapped into one and only one member of the range. Hence, x may not be 2, 6, or 11; these numbers are already first components of ordered pairs in f(x).

One of the following is equivalent to the statement "If Joan sings, then I will play my guitar"?
A. Joan sings and I will play my guitar.
B. Joan does not sing and I will not play my guitar.
C. Joan sings or I will not play my guitar.
D. Joan does not sing or I will play my guitar.
(D) Given a conditional statement in the form "If P then Q," an equivalent statement is in the form "Not P or Q." In this example, P is the statement "Joan sings" and Q is the statement "I will play my guitar."

How many four-digit numbers are there such that the first digit is odd, the second is even, and there is no repetition of digits?
A. 1,200
B. 1,625
C. 200
D. 1,400
(D) There are five possibilities for the first (1, 3, 5, 7, 9), five for the second (0, 2, 4, 6, 8), eight possibilities for the third (10 - 2), and seven for the fourth (8 - 1) = 7. We subtracted 2 possibilities on the third and 3 on the fourth because there is no repetition, as shown below: 5 x 5 x (10 - 2)(8 - 1) = 5 x 5 x 8 x 7 = 1,400

Which one of the following is a valid argument?
A. All rainy days are cloudy.Yesterday was not cloudy.Yesterday was not rainy.
B. All trees have brown leaves.This plant has brown leaves.This plant is a tree.
C. Some wolves are vicious.This animal is vicious.This animal is a wolf.
D. Some people have stocks and bonds.Charles has stocks.Charles has bonds.
(A) An argument is valid if given that the premises are true, then the conclusion must be true. Only answer choice (A) would satisfy this definition. Answer choice (B) is wrong because other objects besides trees may have brown leaves. Answer choice (C) is wrong because animals other than wolves may be vicious. Answer choice (D) is wrong because people may own only stocks, only bonds, both stocks and bonds, or neither stocks nor bonds.

Suppose that point K is reflected across the line y = x. If the original coordinates of K are (–9, 3), what will be K’s new coordinates?
A. (–9, –3)
B. (–3, 9)
C. (3, –9)
D. (9, –3)
(C) When any point (x, y) is reflected across the line y = x, the coordinates are simply reversed so that the new coordinates become (y, x). Thus, (–9, 3) becomes (3, –9).

The function f (x) is defined as follows:

f(x) = 4x-1 , if x ≤ -5 
= 5x + 1 , if x > -5 
What is the value of f (–7) + f (10)?
The correct answer is 22.
f (-7) = (4)(-7) - 1 = -29 and f (10) = (5)(10) + 1 = 51. Then -29 + 51 = 22.

Which one of the following is true for any function?
A. A horizontal line may only intersect the graph of the function once.
B. The inverse must also be a function.
C. A vertical line may only intersect the graph of the function once.
D. The function must be defined for all real numbers.
(C) Since each x value in the domain can correspond only to one y value in the range, any vertical line can intersect the graph at most once. Answer choice (A) is wrong because the same y value may correspond to two different x values. An example to show why answer choice (B) is wrong would be f (x) = {(1, 2), (2, 3), (4, 3)}. f (x) is a function consisting of three points. Its inverse would be {(2, 1), (3, 2), (3, 4)}, which is not a function. An example to show why answer choice (D) is wrong would be g (x) = {(1, 5), (3, 7)}. f (x) is only defined for x = 1 or x = 3.

What is the domain of f(x) = √(7-x) ?
A. All numbers less than or equal to 7
B. All numbers greater than or equal to 7
C. All numbers less than or equal to -7
D. All numbers greater than or equal to -7
(A) The domain is defined as all numbers for which 7 – x is at least zero. Solving 7 – x ≥ 0, we get x ≤ 7.