## Monthly Archives: October 2019

### Set theory, functions, relations: part VI

What follows are some more practice questions on functions. The questions are not challenging but we can say that they do lead to conceptual clarity and present some standard set of questions on this topic (it behooves every beginner in calculus or IITJEE mains or RMO or pre RMO to try these set of questions):

1. Find the domain and range of the function: $f(x)=\frac{x-2}{3-x}$
2. If $f(x)=3x^{3}-5x^{2}+9$, find $f(x-1)$.
3. If $f(x)=x^{3}-\frac{1}{x^{3}}$, show that $f(x)+f(\frac{1}{x})=0$
4. If $f(x)=\frac{x+1}{x-1}$ show that $f(f(x))=x$.
5. Find the domain and range of the real valued function $f(x)=\frac{x^{2}+2x+1}{x^{2}-8x+12}$
6. Find the domain of the real valued function of a real variable: $f(x)=\frac{x-2}{2-x}$
7. Find the domain and range of the real valued function $f(x)=\frac{1}{1-x^{2}}$.
8. A function $f: \Re \longrightarrow \Re$ is defined by $f(x)=\frac{3x}{5}+2$ where $x \in \Re$. Does the inverse of f exist? If so, find it. Also, find the domain and range of the inverse.
9. A function is defined piece-wise as follows: $f(x)=3x+5$ for $- 4 \leq x \leq 0$ and $f(x)=5-3x$ for $0 < x \leq 4$, find $f(f(\frac{5}{2}))$; the domain and range of f; and the value of x for which $f(x)=-4$
10. If $f: \Re \longrightarrow \Re$ and $g: \Re \longrightarrow \Re$ given by $f(x)=x-5$ and $g(x)=x^{2}-1$, find (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ and (d) $g \circ g$
11. Find $f(g(x))$ and $g(f(x))$ if (a) $f(x)=3x-1$ and $g(x)=x^{2}+1$ (b) $f(x)=2x$ and $g(x)=4x+1$
12. If $f(x)=\frac{3x+4}{5x-7}$ and $g(x)=\frac{7x+4}{5x-3}$ prove that $f(g(x))=g(f(x))=x$
13. Find the domain and range of the following functions: (a) $f(x)=x^{2}$ (b) $f(x)=\sqrt{(x-1)(3-x)}$ (c) $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ (d) $f(x)=\frac{x+3}{x-3}$ (e) $f(x)=\sqrt{9-x^{2}}$ (f) $f(x)=\sqrt{\frac{x-2}{3-x}}$
14. Find the range of each of the following functions: (a) $f(x)=3x-4$, when $-1 \leq x <3$ (b) $f(x)=9-2x^{2}$ for $-5 \leq x \leq 3$ (c) $f(x)=x^{2}-6x+11$ for all $x \in \Re$.
15. Solve the following: (a) if $f(x)=\frac{x^{3}+1}{x^{2}+1}$, find $f(-3)$, and $f(-1)$. (b) If $f(x)=(x-1)(2x+1)$, find $f(1)$, $f(2)$, $f(-3)$. (c) If $f(x)=2x^{2}-3x-1$, find $f(x+2)$.
16. Which of the following relations are functions? Justify your answer. If it is a function, determine its range and domain. (a) $\{ (2,1),(4,2),(6,3),(8,4), (10,5), (12,6),(14,7)\}$ (b) $\{ (2,1),(3,1),(5,2)\}$ (c) $\{ (2,3),(3,2),(2,5),(5,2)\}$ (d) $\{ (0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)\}$
17. Find a, if $f(x)=ax+5$, and $f(1)=8$
18. If $f(x)=f(3x-1)$ for $f(x)=x^{2}-4x+11$, find x.
19. If $f(x)=x^{2}-3x+4$, then find the value of x satisfying $f(x)=f(2x+1)$.
20. Let $A = \{ 1,2,3,4 \}$ and $Z$ be the set of integers. Define $f:A \longrightarrow Z$ by $f(x)=3x+7$. Show that f is a function from A to Z. Also, find the range of f.
21. Find if the following functions are one-one or onto or bijective: (a) $f: \Re \longrightarrow \Re$ (b) $f: Z \longrightarrow Z$ given by $f(x)=x^{2}+4$ for all $x \in Z$.
22. Find which of the following functions are surjective, injective or bijective or none of these : (a) $f: \Re \longrightarrow \Re$ as $f(x)=3x+7$ for all $x \in \Re$ (b) $f: \Re \longrightarrow \Re$ given as $f(x)=x^{2}$ for all $x \in \Re$ (c) $f = \{ (1,3),(2,6),(3,9),(4,12)\}$ defined from A to B where $A = \{ 1,2,3,4\}$ and $B = \{ 5,6,9,12,15\}$
23. Let f and g be two real valued functions defined by $f(x)=x+1$ and $g(x)=2x-9$. Find $f+g$ and $f-g$ and $\frac{f}{g}$.
24. Find $g \circ f$ and $f \circ g$ where (a) $f(x)=x-2$ and $g(x)=x^{2}+3x+1$ (b) $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-2}{x+2}$.
25. If $f(x)= \frac{2x+3}{3x-2}$ prove that $f \circ f$ is an identity function.
26. If $f(x)=\frac{3x+2}{4x-1}$ and $g(x)=\frac{x+2}{4x-3}$, prove that $(g \circ f)(x)=(f \circ g)(x)=x$.
27. If $f = \{ (2,4),(3,6),(4,8),(5,10),(6,12)\}$ and $g = \{ (4,13),(6,19),(8,25),(10,31),(12,37)\}$ find $g \circ f$.
28. Show that $f:\Re \longrightarrow \Re$ given by $f(x)=3x-4$ is one-one and onto also. Find its inverse function also. Also, find the domain and range of the inverse function. Also find $f^{-1}(9)$ and $f^{-1}(-2)$
29. Let $f: \Re-\{ 2\} \longrightarrow \Re$ be defined by $f(x)=\frac{x^{2}-4}{x-2}$ and $g: \Re \longrightarrow \Re$ be defined by $g(x)=x+2$. Find whether the two functions f and g are same, or not same. Justify your answers.

Regards,

Nalin Pithwa

### Set theory, relations, functions: preliminaries: Part V

Types of functions: (please plot as many functions as possible from the list below; as suggested in an earlier blog, please use a TI graphing calculator or GeoGebra freeware graphing software):

1. Constant function: A function $f:\Re \longrightarrow \Re$ given by $f(x)=k$, where $k \in \Re$ is a constant. It is a horizontal line on the XY-plane.
2. Identity function: A function $f: \Re \longrightarrow \Re$ given by $f(x)=x$. It maps a real value x back to itself. It is a straight line passing through origin at an angle 45 degrees to the positive X axis.
3. One-one or injective function: If different inputs give rise to different outputs, the function is said to be injective or one-one. That is, if $f: A \longrightarrow B$, where set A is domain and set B is co-domain, if further, $x_{1}, x_{2} \in A$ such that $x_{1} \neq x_{2}$, then it follows that $f(x_{1}) \neq f(x_{2})$. Sometimes, to prove that a function is injective, we can prove the conrapositive statement of the definition also; that is, $y_{1}=y_{2}$ where $y_{1}, y_{2} \in codomain \hspace{0.1in} range$, then it follows that $x_{1}=x_{2}$. It might be easier to prove the contrapositive. It would be illuminating to construct your own pictorial examples of such a function.
4. Onto or surjective: If a function is given by $f: X \longrightarrow Y$ such that $f(X)=Y$, that is, the images of all the elements of the domain is full of set Y. In other words, in such a case, the range is equal to co-domain. it would be illuminating to construct your own pictorial examples of  such a function.
5. Bijective function or one-one onto correspondence: A function which is both one-one and onto is called a bijective function. (It is both injective and surjective). Only a bijective function will have a well-defined inverse function. Think why! This is the reason why inverse circular functions (that is, inverse trigonometric functions have their domains restricted to so-called principal values).
6. Polynomial function: A function of the form $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots + a_{n}x^{n}$, where n is zero or positive integer only and $a_{i} \in \Re$ is called a polynomial with real coefficients. Example. $f(x)=ax^{2}=bx+c$, where $a \neq 0$, $a, b, c \in \Re$ is called a quadratic function in x. (this is a general parabola).
7. Rational function: The function of the type $\frac{f(x)}{g(x)}$, where $g(x) \neq 0$, where $f(x)$ and $g(x)$ are polynomial functions of x, defined in a domain, is called a rational function. Such a function can have asymptotes, a term we define later. Example, $y=f(x)=\frac{1}{x}$, which is a hyperbola with asymptotes X and Y axes.
8. Absolute value function: Let $f: \Re \longrightarrow \Re$ be given by $f(x)=|x|=x$ when $x \geq 0$ and $f(x)=-x$, when $x<0$ for any $x \in \Re$. Note that $|x|=\sqrt{x^{2}}$ since the radical sign indicates positive root of a quantity by convention.
9. Signum function: Let $f: \Re \longrightarrow \Re$ where $f(x)=1$, when $x>0$ and $f(x)=0$ when $x=0$ and $f(x)=-1$ when $x<0$. Such a function is called the signum function. (If you can, discuss the continuity and differentiability of the signum function). Clearly, the domain of this function  is full $\Re$ whereas the range is $\{ -1,0,1\}$.
10. In part III of the blog series, we have already defined the floor function and the ceiling function. Further properties of these functions are summarized (and some with proofs in the following wikipedia links): (once again, if you can, discuss the continuity and differentiablity of the floor and ceiling functions): https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
11. Exponential function: A function $f: \Re \longrightarrow \Re^{+}$ given by $f(x)=a^{x}$ where $a>0$ is called an exponential function. An exponential function is bijective and its inverse is the natural logarithmic function. (the logarithmic function is difficult to define, though; we will consider the details later). PS: Quiz: Which function has a faster growth rate — exponential or a power function ? Consider various parameters.
12. Logarithmic function: Let a be a positive real number with $a \neq 1$. If $a^{y}=x$, where $x \in \Re$, then y is called the logarithm of x with base a and we write it as $y=\ln{x}$. (By the way, the logarithmic function is used in the very much loved mp3 music :-))

Regards,

Nalin Pithwa