Question

Number of integers in the domain of the function $f\left(x\right)={\log }_{(6-x)}(7x-x^2)$ is

• A. 6
• B. 5
• C. 4
• D. infinite

Answer 1

Answer: (C)

4

$f\left(x\right)=lo{g}_{(6-x)}(7x-x^2)$
$=\frac{log(7x-x^2)}{log(6-x)}$
Now
$7x-x^2>0$
$x(7-x)>0$
$x>0$ and $x<7$ ...(i)
And $6-x>0$
Or
$x<6$ ...(ii)
Hence
$xϵ(0,6)$
The integral points will be $\{1,2,3,4,5\}$
However
as $x\to 5$
$log(6-x)\to 0$.
Hence
$f\left(x\right)\to \infty$.
Thus the integral points will be $\{1,2,3,4\}$
Hence 4 integral points.