Determine if Continuous f(x)=(x^2-8x+15)/(x^2-9)
The question asks whether the function f(x) = (x^2 - 8x + 15) / (x^2 - 9) is continuous. It's a problem of calculus, concerning the properties of continuity of a rational function. Continuity, in this context, generally means that the function does not have any abrupt changes, holes, or jumps in its graph. The function is given as a fraction where both the numerator and the denominator are polynomials. To determine continuity, you typically need to examine points where the denominator might be zero, causing the function to potentially become undefined or have discontinuities.
$f \left(\right. x \left.\right) = \frac{x^{2} - 8 x + 15}{x^{2} - 9}$
Answer
Solution:
Determine the Continuity of the Function $f(x) = \frac{x^2 - 8x + 15}{x^2 - 9}$
Step 1: Identify the Domain for Continuity
Begin by setting the denominator of $\frac{x^2 - 8x + 15}{x^2 - 9}$ to zero to find the points of discontinuity.
$$x^2 - 9 = 0$$
Step 1.1: Solve the Equation for $x$
Add $9$ to both sides to isolate $x^2$.
$$x^2 = 9$$
Extract the square root of both sides to solve for $x$.
$$x = \pm \sqrt{9}$$
Step 1.2: Simplify the Square Root
Express $9$ as a square of an integer.
$$x = \pm \sqrt{3^2}$$
Simplify the radical to obtain the numerical values.
$$x = \pm 3$$
Step 1.3: Determine the Complete Set of Solutions
Consider both the positive and negative square roots.
For the positive root:
$$x = 3$$
For the negative root:
$$x = -3$$
Combine both solutions.
$$x = 3, -3$$
Step 1.4: Establish the Domain
Exclude the values that make the denominator zero.
Interval Notation:
$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$
Set-Builder Notation:
$\{x | x \neq 3, -3\}$
Step 2: Assess Continuity Based on the Domain
Since the domain excludes $x = 3$ and $x = -3$, the function $\frac{x^2 - 8x + 15}{x^2 - 9}$ is not continuous for all real numbers.
- The function is discontinuous.
Knowledge Notes:
To determine the continuity of a function, particularly a rational function like $f(x) = \frac{x^2 - 8x + 15}{x^2 - 9}$, we need to consider the following points:
Domain of the Function: The domain includes all real numbers for which the function is defined. For rational functions, this means identifying values that make the denominator zero since division by zero is undefined.
Discontinuities: Points where the function is not defined (such as where the denominator is zero) are considered discontinuities. A function can have point discontinuities (removable), jump discontinuities, or infinite discontinuities (non-removable).
Continuity Over an Interval: A function is continuous over an interval if it is continuous at every point within that interval. If there are any values within the interval where the function is not defined or not continuous, the function is not continuous over that interval.
Interval Notation: This is a way to represent intervals on the real number line. It uses parentheses to denote open intervals and square brackets for closed intervals. For example, $(a, b)$ represents all numbers between $a$ and $b$, not including $a$ and $b$ themselves.
Set-Builder Notation: This notation describes a set by specifying a property that its members must satisfy. For example, $\{x | x \neq 3, -3\}$ describes all real numbers $x$ such that $x$ is not equal to $3$ or $-3$.
Solving Quadratic Equations: When solving equations like $x^2 = 9$, we take the square root of both sides, which yields two solutions, one positive and one negative, because both $3^2$ and $(-3)^2$ equal $9$.
Function Behavior: Understanding the behavior of a function around its discontinuities is important. For instance, if the function approaches a finite value from both sides of a point of discontinuity, it may have a removable discontinuity, which can be "fixed" by redefining the function at that point. If the function goes to infinity, the discontinuity is non-removable.
By examining these aspects, we can determine the continuity of a function and identify intervals where the function is continuous or discontinuous.
