How do you describe the intervals(s) on which the below function is continuous, using interval notation?$$f(x)=x\sqrt(x+6)$$

Question

Vedantu Content Team · Accepted Answer

Hint: For a function which is a product of two or more functions to be continuous. For example, $$f(x)=g(x)h(x)$$, for $$f(x)$$ to be continuous $$g(x)$$ and $$ h(x)$$ both need to be continuous. So, we can find the range on which $$f(x)$$ is continuous by finding separately the range on which $$g(x)$$ and $$ h(x)$$ both are defined and then taking the intersection of the ranges.Complete step by step answer:The given function is $$f(x)=x\sqrt(x+6)\Rightarrow x\sqrt{x+6}$$. It is of the form of $$f(x)=g(x)h(x)$$. We know that to find the range on which $$f(x)$$ is continuous we need to separately find the range on which $$g(x)$$ and $$ h(x)$$ both are defined, and then we take the intersection of the ranges. We will do the same for this function.Here, $$g(x)=x$$ and $$h(x)=\sqrt{x+6}$$, For the function $$g(x)=x$$, as it is a linear function, it is defined for $$\left( -\infty ,\infty  \right)$$.For the function $$h(x)=\sqrt{x+6}$$, it is of the form $$\sqrt{a}$$. We know that for $$\sqrt{a}$$ to be defined, the only condition required is $$a$$ must be a non-negative quantity. Hence for $$h(x)=\sqrt{x+6}$$, to be defined $$x+6\ge 0$$. Subtracting 6 from both sides of this equation, we get$$ \Rightarrow x+6-6\ge 0-6 $$$$ \Rightarrow x\ge -6 $$So for $$h(x)=\sqrt{x+6}$$ to be defined, the range is $$\left[ -6,\infty  \right)$$.We have range for both functions, we need to take their intersection $$\Rightarrow \left( -\infty ,\infty  \right)\bigcap \left[ -6,\infty  \right)$$$$\Rightarrow \left[ -6,\infty  \right)$$Hence the range on which $$f(x)=x\sqrt(x+6)$$ is continuous is $$\left[ -6,\infty  \right)$$.Note: We can do the same thing for addition, subtraction, or division of the function. That is for functions of the form $$f(x)=g(x)+h(x)$$, $$f(x)=g(x)-h(x)$$, $$f(x)=\dfrac{g(x)}{h(x)}$$. Here for the function of the form $$f(x)=\dfrac{g(x)}{h(x)}$$, the values for which $$h(x)=0$$ should be excluded from the range.

How do you describe the intervals(s) on which the below function is continuous, using interval notation?\[f(x)=x\sqrt(x+6)\]

How do you describe the intervals(s) on which the below function is continuous, using interval notation?
\[f(x)=x\sqrt(x+6)\]