Relations and Functions: Mixed Practice

Before you begin#

Twelve ques­tions on rela­tions and func­tions. Make one habit auto­matic: for every func­tion ques­tion, write down the domain before you do any­thing else. Many wrong answers come from divid­ing by zero or tak­ing the square root of a neg­a­tive with­out notic­ing.

Ques­tions#

1. Find $a$ and $b$ if $(a + 3, 2b - 1) = (5, a + b)$.
2. $A = \{a, b\}$. Write $A \times A$ and count the rela­tions on $A$.
3. $A = \{1, 2, 3\}$, $B = \{2, 3\}$, $C = \{3, 4\}$. Ver­ify $A \times (B \cup C) = (A \times B) \cup (A \times C)$.
4. On $\{1, 2, 3, 4, 5\}$, $R = \{(a, b) : a < b \text{ and } a + b \text{ is odd}\}$. List $R$; give its domain and range.
5. Is $\{(2, 1), (5, 1), (8, 1), (11, 1)\}$ a func­tion? What are its domain and range?
6. $\displaystyle f(x) = \frac{x - 1}{x + 2}$. Find the domain, $f(0)$, and the value of $x$ for which $f(x) = 3$.
7. Find the domain and range of $f(x) = \sqrt{16 - x^2}$.
8. Find the range of $f(x) = \lvert x - 3 \rvert + \lvert x + 1 \rvert$.
9. $f(x) = x + 1$, $g(x) = x^2 - 1$. Find $\displaystyle \left(\tfrac{g}{f}\right)(x)$ and its domain; sim­plify.
10. Draw the graph of the signum func­tion and state its range.
11. Mak­ing $x$ pens costs $C(x) = 300 + 8x$ rupees, and each pen sells for ₹$14$. Write the profit $P(x)$, and find how many pens must be sold to break even.
12. A lin­ear func­tion on $\mathbf{Z}$ con­tains $(0, -2)$ and $(3, 7)$. Find $f(x)$.

Answers#