Question

If $f(x) = \ln ({x^2} + |x| + 10)$ 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

We use the definition of the modulus of x. that is if $x \geqslant 0$then $|x| = x$ and if $x < 0$ then $|x| = - x$. We will find the domain of the given function. By drawing the graph at 0$x \geqslant 0$ and $x < 0$ we will find the minimum value of the function then on further simplification we find the natural domain.

Complete step by step answer:
Given $f(x) = \ln ({x^2} + |x| + 10)$,
Take, ${x^2} + 7|x| + 10 > 0$ (Because it is obvious that it is greater than zero)
When, $x \geqslant 0$ we have ${x^2} + 7x + 10$.
When, $x < 0$ we have ${x^2} - 7x + 10$.
Now solving one by one we have,
$\Rightarrow {x^2} + 7x + 10 > 0$ (It is obvious for the values of x).
Factoring we get,
$\Rightarrow {x^2} + 5x + 2x + 10 > 0$
$\Rightarrow x(x + 5) + 2(x + 5) > 0$
$\Rightarrow (x + 5)(x + 2) > 0$

We can say that $x \in [0,\infty ){\text{ - - - - - - - (1)}}$.
Similarly,
$\Rightarrow {x^2} - 7x + 10 < 0$ (It is obvious for the values of x)
Factoring we get,
$\Rightarrow {x^2} - 5x - 2x + 10 < 0$
$\Rightarrow x(x - 5) - 2(x - 5) < 0$
$\Rightarrow (x - 5)(x - 2) < 0$
Now,

We can tell that, $x \in ( - \infty ,0){\text{ - - - - - (2)}}$
If we take the intersection of (1) and (2) we get the domain $x \in R$.
Now, draw the graph of ${x^2} + 7x + 10$ and ${x^2} - 7x + 10$

We can see that f is minimum at 10. But maximum is infinity.
Now to plot the other,

We can see that f is minimum at 10 and maximum is infinity.
That is $10 \leqslant {x^2} + 7x + 10 < \infty$
Taking logarithm above,
$\ln 10 \leqslant \ln ({x^2} + 7x + 10) < \ln \infty$
$\ln 10 \leqslant f(x) < \infty$
Hence the natural domain is $[\ln 10,\infty )$

Hence the correct option is (B).

Note: If we have $\leqslant$ or $\geqslant$ we take a closed interval. If we have $>$or $<$ we take an open interval. We can draw the graph by putting x values as 0, 1, 2, 3,… we get the value of f(x) which is treated as y values then we can plot the graph to get the minimum values as done above.