Drawing a Graph

Q. Let $f\left(x\right)=x^3-3x+1.$ Then

1. $f\left(f\right(x\left)\right)=0$ has $7$ real solutions
2. $f\left(f\right(x\left)\right)=0$ has $4$ real solutions
3. $f\left(f\right(x\left)\right)=-1$ has $7$ real solutions
4. $f\left(f\right(x\left)\right)=-1$ has $4$ real solutions

Q. $f\left(x\right)=|9-x^2|-|x-a|$ then which of the following statements is/are true?

1. For a = 8, number of distinct real roots of f(x)=0 is 4
2. For a = 8, number of distinct real roots of f(x)=0 is 3
3. For a = 3, number of distinct real roots of f(x)=0 is 4
4. For a = 3, number of distinct real roots of f(x)=0 is 3

Q. The number of distinct real solution of $x^4-4x^3+12x^2+x-1=0$.

Q. Let $p\left(x\right)=x^2+ax+b$ have two distinct real roots, where $a,b$ are real numbers. Define $g\left(x\right)=p\left(x^3\right)$ for all real numbers $x$. Then which the following statements are true?
I. $g$ has exactly two distinct real roots
II. $g$ can have more than two distinct real roots
III. There exists a real number $\alpha$ such that $g\left(x\right)\ge \alpha$ for all real $x$

1. Only I and III
2. Only II
3. Only II and III
4. Only I

Q. Let $f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1},forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$ where [x] denotes greatest integer function and {x} denotes fractional part function, then which of the following is/are true?

1. f(x) is one - one function
2. f(x) is onto but not one - one function
3. f(x) is a bijective function
4. f(x) is symmetric about $x=\frac{5}{4}$

Q. Let $f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1},forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$ where [x] denotes greatest integer function and {x} denotes fractional part function, then which of the following is/are true?

1. f(x) is one - one function
2. f(x) is onto but not one - one function
3. f(x) is a bijective function
4. f(x) is symmetric about $x=\frac{5}{4}$

Q. $f\left(x\right)=|9-x^2|-|x-a|$ then which of the following statements is/are true?

1. For a = 8, number of distinct real roots of f(x)=0 is 4
2. For a = 8, number of distinct real roots of f(x)=0 is 3
3. For a = 3, number of distinct real roots of f(x)=0 is 4
4. For a = 3, number of distinct real roots of f(x)=0 is 3

Q. Let $f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1},forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$ where [x] denotes greatest integer function and {x} denotes fractional part function, then which of the following is/are true?

1. f(x) is one - one function
2. f(x) is onto but not one - one function
3. f(x) is a bijective function
4. f(x) is symmetric about $x=\frac{5}{4}$

Q. $f\left(x\right)=|9-x^2|-|x-a|$ then which of the following statements is/are true?

1. For a = 8, number of distinct real roots of f(x)=0 is 4
2. For a = 8, number of distinct real roots of f(x)=0 is 3
3. For a = 3, number of distinct real roots of f(x)=0 is 4
4. For a = 3, number of distinct real roots of f(x)=0 is 3

Q. Let $f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1},forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$ where [x] denotes greatest integer function and {x} denotes fractional part function, then which of the following is/are true?

1. f(x) is one - one function
2. f(x) is onto but not one - one function
3. f(x) is a bijective function
4. f(x) is symmetric about $x=\frac{5}{4}$