Question

If \[f(x) = \ln ({x^2} + 7\left| x \right| + 10)\] is a single valued real function then the range of\[f(x)\] in its natural domain will be
A) \[[0, + \infty ]\]
B) \[[\ln 10, + \infty ]\]
C) \[[0,10]\]
D) \[R\]

Answer 1

Hint:
Here we will form a condition as the log function is present in the given equation. Then we will analyze and solve the equation. Simplifying the equation further we will get the natural domain of the given function\[f(x)\].

Complete step by step solution:
The given function is\[f(x) = \ln ({x^2} + 7\left| x \right| + 10)\]
Firstly we will form a condition as the log function is present. So, from the given expression we can say that the log term is positive as log can only take the positive value. Therefore, we get
\[{x^2} + 7\left| x \right| + 10 > 0\]………………. \[\left( 1 \right)\]
 This condition is valid for both the positive and negative value of the variable \[x\].
Now from the equation \[\left( 1 \right)\] we can see that it have a term of
\[{x^2}\]. So, whatever the value of\[x\] is i.e. positive or negative. The value of this term i.e. \[{x^2}\] is always positive.
Similarly from the equation \[\left( 1 \right)\] we can see that it have a term of\[7\left| x \right|\]. So, whatever the value of \[x\] is i.e. positive or negative. The value of this term i.e. \[7\left| x \right|\] is always positive because the variable \[x\] is in the mod function.
So by this we came to know that the value of the expression \[{x^2} + 7\left| x \right| + 10\] is always positive.
So the upper limit of the function is positive infinity i.e. \[ + \infty \].
Now we have to find out the lower limit of the function. So, by putting the value of\[x = 0\] in the equation \[\left( 1 \right)\] we will get the lower limit of the function. Therefore, we get
\[ \Rightarrow {x^2} + 7\left| x \right| + 10 = {0^2} + 7 \times \left| 0 \right| + 10 = 10\]
So, \[ \Rightarrow {x^2} + 7\left| x \right| + 10 \ge 10\]
Applying natural log to both sides, we get
\[ \Rightarrow \ln \left( {{x^2} + 7\left| x \right| + 10} \right) \ge \ln \left( {10} \right)\]
\[ \Rightarrow f(x) \ge \ln \left( {10} \right)\]
So,\[\ln \left( {10} \right)\] is the lower limit of the function and \[ + \infty \] is the upper limit of the function.
 Hence, the natural domain of the function\[f(x)\] is \[[\ln 10, + \infty ]\]

So, option B is the correct option.

Note:
We should know that the natural domain of a function is the range of the function where its value can lie. Also we should know that the value of any term inside the log function is always positive; it can never be negative. Also we have to keep in mind the type of bracket which should be used to show the natural domain of the function. Either the brackets may be open bracket or closed bracket.
Open bracket is used to show the natural domain of the function when the end points are not included in it. For example: \[(2,15)\] which means \[2 < x < 15\].
Closed bracket is used to show the natural domain of the function when the end points are included in it. For example: \[[2,15]\] which means \[2 \le x \le 15\].