Find the Range of \(f(x)=\sqrt{1-x^2}\)

Solution

Given

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

Step 1: Find the Domain

Since the quantity inside the square root must be non-negative,

\[ 1-x^2\ge0 \] \[ x^2\le1 \] \[ -1\le x\le1 \]

Step 2: Find the Range

Let

\[ y=\sqrt{1-x^2} \]

Since square root always gives non-negative values,

\[ y\ge0 \]

Also,

\[ 1-x^2\le1 \]

Therefore,

\[ y\le1 \]

When

\[ x=0, \] \[ y=1 \]

and when

\[ x=\pm1, \] \[ y=0 \]

Hence all values from \(0\) to \(1\) are attained.