# Domain of One Variable Function – A function with polynom inside a square root – Exercise 2421

Exercise

Determine the domain of the function:

$$f(x)=\sqrt{1-x^2}$$

$$-1\leq x\leq 1$$

Solution

Let’s find the domain of the function:

$$f(x)=\sqrt{1-x^2}$$

There is a square root, so we need the expression inside the root to be non-negative:

$$1-x^2\geq 0$$

It is a square inequality. The roots of the quadratic equation:

$$1-x^2=0$$

are

$$x_1=1, x_2=-1$$

Also, the coefficient of the square expression is negative (-1), so the graph looks like an inverted parabola (= inverted bowl = “crying”).

It looks like this:

Back to inequality:

$$1-x^2\geq 0$$

We need to check when the equation we solved is not negative, i.e. when the graph is not below the x-axis. And one can see from the graph that this happens when the following holds

$$-1\leq x\leq 1$$

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