Determine if Continuous f(x)=(x^2-5x+4)/(x^2-1)
This question is asking about the continuity of a function that is given in the form of a rational expression (i.e., a fraction where both the numerator and the denominator are polynomials). The function f(x) is specifically (x^2-5x+4)/(x^2-1). The task is to analyze and conclude whether the function f(x) is continuous for all values of x in its domain. The continuity of a function typically means that there are no breaks, jumps, or holes in its graph. In the context of rational functions like this, special attention should be paid to points where the denominator is zero, since these points could potentially be locations of discontinuities.
$f \left(\right. x \left.\right) = \frac{x^{2} - 5 x + 4}{x^{2} - 1}$
Answer
Solution:
Step:1
Identify the range of values for which the function is defined to assess its continuity.
Step:1.1
To find the values that cause the function to be undefined, equate the denominator of $\frac{x^{2} - 5x + 4}{x^{2} - 1}$ to zero: $x^{2} - 1 = 0$.
Step:1.2
Proceed to isolate $x$.
Step:1.2.1
By adding $1$ to each side, we get $x^{2} = 1$.
Step:1.2.2
Extract the square root on both sides to remove the squared term: $x = \pm \sqrt{1}$.
Step:1.2.3
Since the square root of $1$ is $1$, we have $x = \pm 1$.
Step:1.2.4
Combine the positive and negative solutions to complete the set of solutions.
Step:1.2.4.1
Using the positive sign from $\pm$, we find the first solution: $x = 1$.
Step:1.2.4.2
Using the negative sign from $\pm$, we find the second solution: $x = -1$.
Step:1.2.4.3
The full set of solutions includes both the positive and negative values: $x = 1, -1$.
Step:1.3
The domain consists of all $x$ values that keep the function defined. In interval notation, this is: $\left(-\infty, -1\right) \cup \left(-1, 1\right) \cup \left(1, \infty\right)$. In set-builder notation, it is: $\{x | x \neq 1, -1\}$.
Step:2
Given that the domain excludes certain real numbers, the function $\frac{x^{2} - 5x + 4}{x^{2} - 1}$ is not continuous across the entire set of real numbers.
Step:3
Knowledge Notes:
To determine if a function $f(x)$ is continuous, we need to ensure that it is defined for all real numbers without any breaks or holes. A function can be discontinuous if there are points where it is not defined, typically where the denominator of a fraction equals zero, causing a division by zero.
Domain of a Function: The set of all possible input values (x-values) for which the function is defined.
Continuity: A function is continuous at a point if the limit as it approaches the point from both sides is equal to the function's value at that point. A function is continuous over an interval if it is continuous at every point within that interval.
Interval Notation: A way to represent intervals on the real number line. For example, $(a, b)$ represents all numbers greater than $a$ and less than $b$. Square brackets, e.g., $[a, b]$, include the endpoints, while parentheses, e.g., $(a, b)$, do not.
Set-Builder Notation: A notation for describing a set by stating the properties that its members must satisfy. For example, $\{x | x \neq 1, -1\}$ describes all real numbers except $1$ and $-1$.
Solving Quadratic Equations: A quadratic equation in the form $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or using the quadratic formula. In this case, $x^2 - 1 = 0$ is a difference of squares and can be factored as $(x + 1)(x - 1) = 0$.
Square Roots: The square root of a number $a$, denoted $\sqrt{a}$, is a number that, when multiplied by itself, gives $a$. For positive real numbers, there are always two square roots: one positive and one negative.
By analyzing the domain and the continuity of the function, we can conclude whether the function is continuous for all real numbers or if there are points of discontinuity.
