Find the Domain y=( square root of 1-x^2)/x

Solution:

Step 1:

Determine where the function $\sqrt{1-x^2}$ is real by setting the inside of the square root greater than or equal to zero: $1-x^2\ge 0$.

Step 2:

Isolate $x$ in the inequality.

Step 2.1:

Subtract $1$ from both sides: $-x^2\ge -1$.

Step 2.2:

Divide by $-1$, remembering to reverse the inequality: $x^2\le 1$.

Step 2.3:

Extract the square root on both sides: $|x|\le \sqrt{1}$.

Step 2.4:

Simplify the inequality: $|x|\le 1$.

Step 2.5:

Express the inequality without absolute value by considering both cases for $x$.

Step 2.5.1:

For $x\ge 0$: $x\le 1$.

Step 2.5.2:

For $x<0$: $-x\le 1$.

Step 2.6:

Combine the intervals from both cases: $-1\le x\le 1$.

Step 3:

Identify where the denominator $x$ causes the expression to be undefined: $x\ne 0$.

Step 4:

Combine the results to find the domain, excluding the point where the denominator is zero.

Interval Notation: $[-1,0)\cup (0,1]$.

Set-Builder Notation: $\{x|-1\le x\le 1,x\ne 0\}$.

Knowledge Notes:

1. Domain of a Function: The domain of a function is the set of all possible input values (usually $x$ values) for which the function is defined. For real-valued functions, this means excluding values that lead to division by zero, negative square roots, and logarithms of non-positive numbers.

2. Inequalities: When solving inequalities, if you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

3. Absolute Value: The absolute value of a number is its distance from zero on the number line, without considering direction. The inequality $|x|\le a$ can be written as $-a\le x\le a$.

4. Interval Notation: Interval notation is a way of writing subsets of the real number line. A closed interval, such as $[a,b]$, includes the endpoints $a$ and $b$. An open interval, such as $(a,b)$, does not include the endpoints.

5. Set-Builder Notation: Set-builder notation is a concise way of expressing a set by specifying a property that its members must satisfy. For example, $\{x|-1\le x\le 1,x\ne 0\}$ describes the set of all $x$ such that $x$ is between $-1$ and $1$ but not equal to $0$.

6. Square Roots and Radicals: The square root of a number $a$ is a number $b$ such that $b^2=a$. The expression $\sqrt{a}$ is defined only for $a\ge 0$ in the real number system.

7. Combining Intervals: When combining intervals, the union of intervals includes all the values that belong to either interval. The intersection includes only the values that belong to both intervals.