Question

The range of the function ${\log _3}(5 + 4x - {x^2})$ is
a. $(0,2]$
b. $( - \infty ,2]$
c. $(0,9]$
d. None of these

Answer 1

To find the range of the given function we need to first simplify the given function.
Then we need to find the limit to which the function lies.
After that, we will find the other coordinate point to find its range.
We can also find the range with the help of a graph by drawing the curve.
Graph method also helps us to find the $y$ coordinate.

Complete step by step answer:
It is given that ${\log _3}(5 + 4x - {x^2})$ . We aim to find the range of the given function.
Let us consider, $5 + 4x - {x^2}$
Now let us simplify it. Let us take the minus sign commonly.
$5 + 4x - {x^2} = 5 - ({x^2} - 4x)$
Now let us add and subtract four to that.
$= 5 - ({x^2} - 4x + 4 - 4)$
$= 5 - (({x^2} - 4x + 4) - 4)$
Now that can be written as, $5 - ({(x + 2)^2} - 4)$
Now let us simplify it further.
$= 9 - {(x + 2)^2}$
Now we have simplified the function $5 + 4x - {x^2}$ to $9 - {(x + 2)^2}$
Now let us find the limit points of this simplified function. We can see that some value is subtracted from nine. Thus, it should be less than or equal to nine (since the part ${(x + 2)^2}$ can also be zero).
$9 - {(x + 2)^2} \leqslant 9$
Since we cannot determine the negative limit point we take it as $- \infty$ .
Therefore, $- \infty > 9 - {(x + 2)^2} \leqslant 9$
Now let us substitute $t = 5 + 4x - {x^2}$ in the given function we get ${\log _3}(t)$ .
Therefore, $t \in ( - \infty ,9]$
Since $\log$ cannot take negative values, we will rewrite the limits as $t \in (0,9]$ .
Now let us find the $y$ coordinate point for this limit. When a curve is drawn for this function, the curve will meet the point $x = 9$ at $y = 2$ . Thus, the range will be $( - \infty ,2]$ .
Now let us see the options, option (a) $(0,2]$ cannot be the correct answer since the left limit point stops at zero.
Option (b) $( - \infty ,2]$ is the correct answer since $( - \infty ,2]$ is the range we got in our calculation.
Option (c) $(0,9]$ cannot be the correct answer since it has the right limit of nine.
Option (d) None of these, this cannot be the correct answer since we have got option (b) as the correct answer.

So, the correct answer is “Option b”.

Note: Whenever we need to find the range, we need to first try to determine the limit point of the given function.
With the help of the limit, we can find the range of the given function.
If we cannot find any one of the limit points it will extend to infinity.