Question

How do you determine the domain of $f(x)=\dfrac{1}{\sqrt{x+2}}?$

Answer 1

Hint: Here we have to find the domain and range for the function, first we have to determine the set of values for which the function is defined and determine the set of values which we have obtained from this. As an example for better understand we have $f(x)=\sqrt{x},x>0$ Hence the domain of $f(x)$ is $\left[ 0,+\infty \right]$ Also $f(0)=0$ and $f(x)$ has no finite upper bound. Therefore, the range is also $\left[ 0,+\infty \right]$ for $f(x).$

Complete step by step solution:
The given function we have,
$f(x)=\dfrac{1}{\sqrt{x+2}}$
We have to determine
The domain of the given function.
So,
The domain of a function is the set of all the real value of $x$ input for which the function is defined.
The domain can also be defined as a set of all the real numbers except those that make the function defined.
Now in the case:
$\dfrac{1}{\sqrt{x+2}}$
We can’t have zero for the denominator nor a negative square root since both will make the function undefined, then
$D=x+2>0\Rightarrow $or
$x>-2\Rightarrow $ or
$\left( -2,\infty \right)$

Hence, The domain of function $\left[ -2,\infty \right]$

Additional Information:
We can say that domain is the complete set of possible values of the independent variables for the range, it is the functional in which the complete set of all possible resulting values of dependent variables after substituting the domain. We have to find the domain for each function by looking for the value of the independent variable which we are allowed to use (usually $x)$ for range of function it is a spread of possible $y$ values. Make sure you look for $y$ values for minimum and maximum.

Note: While finding the domain always remember that under square root the value of number must be positive and other is in the fraction problem denominator cannot be zero. Whenever we solve a square root problem many students think that we don't get two answers, that is one positive and one negative. So get confused a square root has one value, not two. Another way for finding domain and range is by using graphs because the domain is set of possibilities input value then the domain of graph is on $x$-axis. Similarly the range is the output value on $y$-axis.