Determine if Continuous f(x)=(x^2-7x+12)/(x^2-9)
Problem Overview:
The task is to assess whether the given function f(x) = (x^2 - 7x + 12) / (x^2 - 9) is continuous. Continuity of a function at a point means that the function is smooth and uninterrupted at that point. To be more specific, the function needs to fulfill three conditions for continuity at any point $a$: The function must be defined at $a$, the limit as x approaches $a$from both sides must exist, and the limit of the function as x approaches $a$must equal the function value at $a$.
Brief Explanation of the Question:
To solve this problem, you would typically check the conditions for continuity at all points in the domain of the function. Special attention should be given to points where the denominator is zero, as these are potential points of discontinuity. The function's domain and points where the function could potentially be undefined or possess limits that do not match the function value should be examined in detail. In this case, one would look closely at the behavior of the function around x = ±3, where the denominator x^2 - 9 equals zero, to determine if it causes discontinuities or if they can be reconciled via limit analysis or simplification.
$f \left(\right. x \left.\right) = \frac{x^{2} - 7 x + 12}{x^{2} - 9}$
Answer
Solution:
Step:1
Identify the domain to check the continuity of the function.
Step:1.1
To find the discontinuities, equate the denominator of $\frac{x^{2} - 7 x + 12}{x^{2} - 9}$ to zero: $x^{2} - 9 = 0$
Step:1.2
Find the values of $x$ that satisfy the equation.
Step:1.2.1
Add $9$ to both sides: $x^{2} = 9$
Step:1.2.2
Extract the square root on both sides: $x = \pm \sqrt{9}$
Step:1.2.3
Simplify the square root.
Step:1.2.3.1
Express $9$ as a square of $3$: $x = \pm \sqrt{3^{2}}$
Step:1.2.3.2
Remove the square root by taking out the square term: $x = \pm 3$
Step:1.2.4
Combine both the positive and negative solutions.
Step:1.2.4.1
Consider the positive square root for the first solution: $x = 3$
Step:1.2.4.2
Consider the negative square root for the second solution: $x = -3$
Step:1.2.4.3
The full solution set includes both solutions: $x = 3 , -3$
Step:1.3
The domain consists of all $x$ values that do not make the expression undefined.
In Interval Notation: $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$ In Set-Builder Notation: $\{ x | x \neq 3, -3 \}$
Step:2
The function $\frac{x^{2} - 7 x + 12}{x^{2} - 9}$ is not continuous for all real numbers due to the domain restrictions.
Step:3
Knowledge Notes:
To determine if a function $f(x)$ is continuous, one must first identify the domain of $f(x)$. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, like the one given $\frac{x^{2} - 7 x + 12}{x^{2} - 9}$, the function is undefined where the denominator is zero, as division by zero is not allowed.
The process of finding the domain involves setting the denominator equal to zero and solving for $x$. This typically results in a polynomial equation, which can be solved using various algebraic techniques such as factoring, applying the quadratic formula, or taking square roots.
Once the values of $x$ that make the denominator zero are found, these are the points of discontinuity. The domain is then all real numbers except these points. The function is continuous at every point in its domain; however, if the domain does not include all real numbers, the function is not continuous over the entire real number line.
In interval notation, the domain is expressed as a union of intervals that exclude the points of discontinuity. In set-builder notation, the domain is expressed as a set of all $x$ such that $x$ is not equal to the points of discontinuity.
In the given problem, the denominator is zero when $x = 3$ or $x = -3$, so the function is not continuous at these points. Thus, the function is not continuous over the entire real number line.
