Questions covered in Unit Test: Sets and Functions (Advanced)

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. If $A = \{1, 2, \{3, 4\}\}$, then the number of proper subsets of $A$ is
2. The domain of the function $f(x) = \frac{1}{\sqrt{|x|-x}}$ is
3. If $A$ and $B$ are two sets such that $n(A) = 115$, $n(B) = 326$, $n(A-B) = 47$, then $n(A \cup B)$ is
4. The range of the function $f(x) = \frac{x^2}{1+x^2}$ is
5. Let $R = \{(x, y) : x, y \in Z, x^2 + y^2 \le 4\}$. The number of elements in the domain of $R$ is
6. Find the domain of the function $f(x) = \sqrt{1-x} + \sqrt{x-3}$.
7. If $f(x) = x^2 - 3x + 1$, find $x \in R$ such that $f(2x) = f(x)$.
8. Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be a relation on $A$ defined by $R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$. Write $R$ in roster form.
9. In a survey of 400 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.
10. Find the domain and range of the real function $f(x) = \frac{1}{1 - 2 \cos x}$.
11. Show that $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ using properties of sets.
12. Case Study: A function $f: R \to R$ is defined as $f(x) = 2x + 1$ if $x > 2$, $x^2$ if $-1 \le x \le 2$, and $3x - 1$ if $x < -1$.
    (i) Find $f(3)$.
    (ii) Find $f(0)$.
    (iii) Find $f(-2)$.
    (iv) Find $f(2)$.