Determine if Continuous f(x)=(x-9)/(x^2-81)
The question is asking to analyze the continuous nature of the function f(x) = (x-9)/(x^2-81). Continuous functions have no breaks, jumps, or holes in their graph within a given domain. The task involves examining the function across its entire domain to check for any points where continuity may be broken. This typically involves looking for values that make the denominator equal to zero, which would lead to undefined points, and points where the limit of the function does not match the function's value.
$f \left(\right. x \left.\right) = \frac{x - 9}{x^{2} - 81}$
Answer
Solution:
Determine the Continuity of the Function f(x) = (x-9)/(x^2-81)
Step:1
Identify the domain to check the continuity of the function.
Step:1.1
To find the discontinuities, equate the denominator of $\frac{x - 9}{x^{2} - 81}$ to zero: $x^{2} - 81 = 0$.
Step:1.2
Find the values of $x$ that satisfy the equation.
Step:1.2.1
Add $81$ to each side to isolate $x^{2}$: $x^{2} = 81$.
Step:1.2.2
Extract the square root from both sides to solve for $x$: $x = \pm \sqrt{81}$.
Step:1.2.3
Simplify the square root of $81$.
Step:1.2.3.1
Express $81$ as a square of $9$: $x = \pm \sqrt{9^{2}}$.
Step:1.2.3.2
Deduce the square root, considering only the principal square root: $x = \pm 9$.
Step:1.2.4
Combine the positive and negative solutions to get the full solution set.
Step:1.2.4.1
Utilize the positive part of $\pm$ to find one solution: $x = 9$.
Step:1.2.4.2
Use the negative part of $\pm$ to find the other solution: $x = -9$.
Step:1.2.4.3
The full solution includes both the positive and negative solutions: $x = 9, -9$.
Step:1.3
The domain consists of all $x$ values that keep the function defined.
In Interval Notation: $(-\infty, -9) \cup (-9, 9) \cup (9, \infty)$.
In Set-Builder Notation: $\{x | x \neq 9, -9\}$.
Step:2
Given that the domain excludes some real numbers, the function $\frac{x - 9}{x^{2} - 81}$ is not continuous across the entire set of real numbers.
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 rational functions, the domain excludes values that make the denominator zero.
Continuity: A function is continuous at a point if the limit of the function as it approaches the point from both directions equals the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.
Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials. The domain of a rational function includes all real numbers except those that cause the denominator to be zero.
Solving Quadratic Equations: To solve a quadratic equation of the form $ax^2 + bx + c = 0$, one can use various methods such as factoring, completing the square, or using the quadratic formula. In this case, the equation $x^2 = 81$ is a simple quadratic equation that can be solved by taking the square root of both sides.
Interval Notation: This is a way of writing subsets of the real number line. An interval notation uses parentheses to indicate that an endpoint is not included, known as an open interval, and square brackets to indicate that an endpoint is included, known as a closed interval.
Set-Builder Notation: This is another way to describe a set, using a rule to define the members of the set. It typically includes a variable, a condition that the members must satisfy, and curly braces to enclose the notation.
