Q1. What is the value of \( \tan(\tan^{-1}(3)) \)?
Solution:
By definition, \( \tan(\tan^{-1}(x)) = x \). Therefore, \( \tan(\tan^{-1}(3)) = 3 \).
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Q2. If h(x) = x^3 - 3x + 2, what is the critical point?
Solution:
h'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so x = 1 and x = -1 are critical points.
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Q3. If C = {1, 2}, what is the number of proper subsets of C?
Solution:
The proper subsets of a set exclude the set itself. For C = {1, 2}, the proper subsets are {∅}, {1}, and {2}, totaling 3 subsets.
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Q4. How many subsets of the set H = {x, y} are there that do not contain the element y?
Solution:
The only subset that does not contain y is {∅} and {x}. Total = 2.
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Q5. If R is a relation defined on set A = {1, 2, 3} such that R = {(1, 2), (2, 3)}, is R a function?
Solution:
A relation is a function if every element in the domain (set A) maps to exactly one element in the codomain. Here, 1 maps to 2, and 2 maps to 3, but 3 has no mapping, so R is not a function.
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Q6. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(a, b) | a < b}. How many ordered pairs are in R?
Solution:
The pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
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Q7. If R is a relation on the set {1, 2, 3, 4} defined by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}, what type of relation is R?
Solution:
R is reflexive because it contains all pairs (a, a) and symmetric because (1,2) implies (2,1).
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Q8. If f(x) = x^2 + 2x + 1, what is the vertex of the parabola?
Solution:
The vertex form is f(x) = (x + 1)^2, so the vertex is (-1, 0).
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Q9. What is the domain of the function f(x) = 1/(x-3)?
Solution:
The function is undefined at x = 3, so the domain is x ≠ 3.
