Determine if Continuous f(x)=(x^2-1)/(x^2-6x+5)
The question provided asks us to analyze the continuity of the given function f(x) = (x^2 - 1) / (x^2 - 6x + 5). The task involves determining whether the function is continuous for all values of x, or if there are any points where the function is not continuous, which would be points where the function has a discontinuity such as a hole, jump, or infinite discontinuity. This usually involves finding the values of x (if any) that make the denominator equal to zero, which are points of potential discontinuities, and then examining the behavior of the function around those points.
$f \left(\right. x \left.\right) = \frac{x^{2} - 1}{x^{2} - 6 x + 5}$
Answer
Solution:
Step:1
Identify the domain to check for continuity of the function.
Step:1.1
To find the domain, set the denominator of $\frac{x^{2} - 1}{x^{2} - 6x + 5}$ to zero: $x^{2} - 6x + 5 = 0$.
Step:1.2
Determine the values of $x$ that satisfy the equation.
Step:1.2.1
Apply the AC method to factor $x^{2} - 6x + 5$.
Step:1.2.1.1
Look for two numbers that multiply to $c$ and add up to $b$, where $c = 5$ and $b = -6$. The numbers are $-5$ and $-1$.
Step:1.2.1.2
Factor the expression into $(x - 5)(x - 1) = 0$.
Step:1.2.2
Recognize that if either factor equals zero, the equation is satisfied: $x - 5 = 0$ or $x - 1 = 0$.
Step:1.2.3
Solve $x - 5 = 0$ for $x$.
Step:1.2.3.1
Isolate $x$ by setting $x - 5$ to zero: $x - 5 = 0$.
Step:1.2.3.2
Add $5$ to both sides to find $x = 5$.
Step:1.2.4
Solve $x - 1 = 0$ for $x$.
Step:1.2.4.1
Isolate $x$ by setting $x - 1$ to zero: $x - 1 = 0$.
Step:1.2.4.2
Add $1$ to both sides to find $x = 1$.
Step:1.2.5
The solutions are the values of $x$ that satisfy $(x - 5)(x - 1) = 0$: $x = 5$ and $x = 1$.
Step:1.3
The domain consists of all $x$ values that do not make the denominator zero. In interval notation: $(-\infty, 1) \cup (1, 5) \cup (5, \infty)$. In set-builder notation: $\{x | x \neq 1, 5\}$.
Step:2
The function $\frac{x^{2} - 1}{x^{2} - 6x + 5}$ 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, we need to ensure that it is defined for all $x$ in its domain, and that there are no breaks, holes, or jumps in its graph. For rational functions, such as the one given, continuity is primarily concerned with the points where the denominator is zero, as these are the points where the function is not defined.
The process involves:
Finding the Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a rational function, the domain excludes values that make the denominator zero.
Factoring: To find these values, we factor the denominator and set it equal to zero. Factoring is the process of breaking down a complex expression into simpler ones that, when multiplied together, give the original expression.
Solving Equations: Once factored, we solve for the roots of the equation, which are the x-values that make the denominator zero.
Interval and Set-Builder Notation: The domain is then expressed in interval notation, which provides ranges of x-values, and set-builder notation, which defines a set by a property that its members must satisfy.
Continuity: A function is continuous if it is defined at every point in its domain and does not have any breaks, holes, or jumps. If the domain of a function excludes any real numbers (as it does in this case), the function is not continuous over all real numbers.
In this problem, the domain of the function $f(x) = \frac{x^{2} - 1}{x^{2} - 6x + 5}$ does not include $x = 1$ and $x = 5$, thus the function is not continuous over the entire set of real numbers.
