Functions Questions for CAT

Let $A=\{a,b,c,d,e,f,g\}$ and $B=\{x,y\}$. The number of functions $f:A\to B$ such that both $x$ and $y$ are used at least once in the mapping is:

If $\log_{10} a = b$, find the value of $10^{3b}$ in terms of $a$.

Range of $f(x)=(x-2)^2+3$.

Total number of roots of $f(|x|)=0$ is:

On which date in 2012 do the prices of cabbage (150+0.1n) and onion (140+0.15n) become equal?

Find the value of $x$ in the equation: $\log_{0.5} x = 25$.

Given that $f(x+5)+f(x-5)=2f(x)$, what is the value of $f(x-10)+f(x+10)$?

Given that $x^3 + y^3 + z^3 = 177$ and $xyz = 59$, where $x$, $y$ and $z$ are real numbers that are not all equal, calculate $\left(\dfrac{x^2+y^2+z^2}{xy+yz+zx}\right)^4$.

Determine the number of non-negative roots for the equation: $3^m - (2m+1) = 0$

Given that $f(x)=\frac{1}{1+x}$, determine the value of $f(f(f(x)))$ when $x=5$.

For all natural numbers $n>3$, define $F(n)=\tfrac{n}{n-3}$ and $G(n)=F(4)F(5)...F(n)$. How many natural numbers $3<n<100$ make $G(n)$ a natural number?

Given that $f(x)=3^x$ and $g(x)=\log_9 x$, what is the result of $gofogof(8)$?

Value of the expression $\left(x^{2}-x+1\right) /(x-1)$ cannot lie between

Given the functions $f(x)=\max(x^2-5,\;2x+3)$, $g(x)=\max(x^2-3,\;2x+5)$, and $h(x)=\min(f(x),g(x))$, what is the minimum value of $h(x)$?

Determine the number of real values of $k$ that satisfy the equation $5\{k\} + 4k = 6[k] + 3$, where $[k]$ represents the greatest integer less than or equal to $k$ and $\{k\}$ denotes the fractional part of $k$.

If for all integers $x$, $f(x)+f(x-1)=4$ and $g(x)=x^2$, and if $f(x^2-x)=6$, then find $f(g(6))+g(f(6))$.

If $A=\{1,2,3,4,5,6\}$ and $B=\{p,q,r,s\}$, then the number of onto functions $f:A\to B$ is:

Let $f(x)$ be the quotient when $5x^4−3x^3+2x^2−1$ is divided by $(x^2+4)$. Let $g(x)$ be the remainder when $2x^3−x+1$ is divided by $(x^2+x+1)$. Find the remainder when $f(x)$ is divided by $g(x)$.

Find the number of integral solutions of $\frac{x+15}{x-1}-\frac{8}{x-1}+\frac{x+24}{x-1}-\frac{10}{x-1}=1.$

Given that $|P-Q|=|R|$ and $|P-R|=|Q|$, determine the value of $P^2-Q^2-R^2$.