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

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