Find the Domain y=( square root of 1-x^2)/x
The problem is asking to determine the set of all real numbers for which the given function, \( y = \frac{\sqrt{1-x^2}}{x} \), is defined. This involves finding the values of ‘x’ such that the square root is defined (i.e., the expression within the square root, \( 1-x^2 \), is greater than or equal to zero) and ensuring that the denominator 'x' is not zero, as division by zero is undefined. The combination of these two conditions will provide the domain of the function.
$y = \frac{\sqrt{1 - x^{2}}}{x}$
Answer
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: $1 - x^2 \geq 0$.
Step 2:
Isolate $x$ in the inequality.
Step 2.1:
Subtract $1$ from both sides: $-x^2 \geq -1$.
Step 2.2:
Divide by $-1$, remembering to reverse the inequality: $x^2 \leq 1$.
Step 2.3:
Extract the square root on both sides: $|x| \leq \sqrt{1}$.
Step 2.4:
Simplify the inequality: $|x| \leq 1$.
Step 2.5:
Express the inequality without absolute value by considering both cases for $x$.
Step 2.5.1:
For $x \geq 0$: $x \leq 1$.
Step 2.5.2:
For $x < 0$: $-x \leq 1$.
Step 2.6:
Combine the intervals from both cases: $-1 \leq x \leq 1$.
Step 3:
Identify where the denominator $x$ causes the expression to be undefined: $x \neq 0$.
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: $\{x | -1 \leq x \leq 1, x \neq 0\}$.
Knowledge Notes:
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.
Inequalities: When solving inequalities, if you multiply or divide by a negative number, you must reverse the direction of the inequality sign.
Absolute Value: The absolute value of a number is its distance from zero on the number line, without considering direction. The inequality $|x| \leq a$ can be written as $-a \leq x \leq a$.
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.
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, $\{x | -1 \leq x \leq 1, x \neq 0\}$ describes the set of all $x$ such that $x$ is between $-1$ and $1$ but not equal to $0$.
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 $a \geq 0$ in the real number system.
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.
