The answer is:
The domain for the function is all the real numbers,
Domain:(-∞,∞)
<h2>Why?</h2>Since we are working with fractions, the only restriction that we will have for the function is when the denominator of the function tends to 0.
We are given the function:
Where, the denominator is given by the expression:
For the given expression (quadratic equation), we have that:
Calculating the discriminat of the quadratic function, in order to know if the denominator of the function has roots (zeroes) at the real numbers, we have:
Now, as we know, if the discriminant of a quadratic function is less than 0, the quadratic function has no roots in the real numbers.
Therefore, since the denominator (quadratic function) has no roots in the real numbers, the domain for the function will be equal to all the real numbers.
Domain:(-∞,∞)
Hence, the answer is the third option, the domain for the function is all the real numbers,
Domain:(-∞,∞)
Have a nice day!