Question

How do you find the intervals on which the function is continuous given \[y = \sqrt {5x + 9} \] ?

Answer 1

Hint: Here in this question we have to find the interval for the function. We have to find the range of the $x$ value where the function is continuous. To determine the interval we use the simple arithmetic operations and hence we obtain the required solution for the question.

Complete step by step solution:
In mathematics, an interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Now consider the function \[y = \sqrt {5x + 9} \]. The square root function cannot work with negative inputs. Therefore, we need to think about when the argument becomes negative. So we have
\[5x + 9 < 0\]
Add -9 to the both sides of the equation we have
\[5x + 9 - 9 < 0 - 9\]
On simplifying we have
\[5x < - 9\]
Divide the above equation by 5 we have
\[\dfrac{{5x}}{5} < \dfrac{{ - 9}}{5}\]
On simplifying the above equation we have
\[ \therefore x < - 1.8\]
Therefore, the interval of the function \[y = \sqrt {5x + 9} \] is \[x \in ( - 1.8,\,\infty )\].In this interval the function will be continuous. The value of x should not be less than the value 1.8.

Hence, the interval of the function \[y = \sqrt {5x + 9} \] is \[x \in ( - 1.8,\,\infty )\].

Note: We must know about the table of multiplication and the simple arithmetic operations which are needed for simplification. Since the square root of a function is a positive and it cannot be negative. so by considering the less than inequality we determine the least value where the function is not negative.