Determine if Continuous f(x)=(x^2-5x+4)/(x^2-16)
This problem is asking for an analysis of the continuity of a given function f(x), which is a rational function defined by (x^2 - 5x + 4) / (x^2 - 16). The question requires one to examine whether the function is continuous over its entire domain or if there are any points at which it may not be continuous. This typically involves looking for values of x that cause the denominator to be zero, as those are points where the function is not defined and thus cannot be continuous. Additionally, it may be necessary to check if there are removable discontinuities where the function is not defined, but it might be possible to define it in such a way that it becomes continuous at those points.
$f \left(\right. x \left.\right) = \frac{x^{2} - 5 x + 4}{x^{2} - 16}$
Answer
Solution:
Determine the Continuity of the Function $f(x) = \frac{x^2 - 5x + 4}{x^2 - 16}$
Step 1: Identify the domain where the function is defined.
Step 1.1: Equate the denominator of $\frac{x^2 - 5x + 4}{x^2 - 16}$ to zero to find the points of discontinuity.
$$x^2 - 16 = 0$$
Step 1.2: Find the values of $x$ that satisfy the equation.
Step 1.2.1: Add $16$ to each side to isolate $x^2$.
$$x^2 = 16$$
Step 1.2.2: Extract the square root from both sides to solve for $x$.
$$x = \pm \sqrt{16}$$
Step 1.2.3: Simplify the square root.
Step 1.2.3.1: Express $16$ as a square of an integer.
$$x = \pm \sqrt{4^2}$$
Step 1.2.3.2: Remove the square root by squaring the base.
$$x = \pm 4$$
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.
$$x = 4$$
Step 1.2.4.2: Consider the negative square root for the second solution.
$$x = -4$$
Step 1.2.4.3: The complete set of solutions includes both.
$$x = 4, -4$$
Step 1.3: The domain consists of all $x$ values that keep the function defined.
- Interval Notation: $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$
- Set-Builder Notation: $\{x | x \neq 4, -4\}$
Step 2: As the domain excludes $x = 4$ and $x = -4$, the function $\frac{x^2 - 5x + 4}{x^2 - 16}$ is not continuous for all real numbers.
Step 3:
Knowledge Notes:
The problem involves determining the continuity of a rational function, which is a function that can be expressed as the ratio of two polynomials. The continuity of such a function depends on the points where the denominator is not equal to zero, since division by zero is undefined.
Domain of a Function: The set of all possible input values (usually 'x') for which the function is defined.
Continuity: A function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.
Discontinuity: A point where a function is not continuous. For rational functions, this typically occurs where the denominator is zero.
Interval Notation: A way to represent a set of numbers along a line. For example, $(a, b)$ represents all numbers between 'a' and 'b', not including 'a' and 'b' themselves.
Set-Builder Notation: A way to describe a set by specifying a property that its members must satisfy. For example, $\{x | x \neq a\}$ describes all numbers except 'a'.
Solving Quadratic Equations: To solve an equation of the form $x^2 = a^2$, one can take the square root of both sides, yielding $x = \pm a$.
In this problem, the steps involve finding where the denominator equals zero to identify the points of discontinuity, and then determining the domain of the function. Since there are values that are not included in the domain (where the function is undefined), the function is not continuous over the entire set of real numbers.
