Determine if Continuous f(x)=(x-5)/(x^2-9x+20)
The task here is to analyze the continuity of the function f(x) = (x-5)/(x^2-9x+20). To do this, one would need to check if there are any points at which the function is not defined or where it does not have a limit that matches the function's value. Continuity generally requires that the function is defined at the point in question, that the limit exists as the function approaches that point from both directions, and that the value of the function at that point is equal to the limit. For rational functions like f(x), potential discontinuities can be found where the denominator is zero, as the function would be undefined at those points. Here, the denominator simplifies to (x-4)(x-5), so there may be potential issues with continuity at x=4 and x=5. The question implies to check and explain continuity at these points or possibly throughout its domain.
$f \left(\right. x \left.\right) = \frac{x - 5}{x^{2} - 9 x + 20}$
Answer
Solution:
Step:1
Identify the domain to check for continuity of the function.
Step:1.1
To find the domain, equate the denominator of $\frac{x - 5}{x^{2} - 9x + 20}$ to zero: $x^{2} - 9x + 20 = 0$.
Step:1.2
Proceed to find the roots of the equation.
Step:1.2.1
Employ the AC method to factorize $x^{2} - 9x + 20$.
Step:1.2.1.1
In the quadratic form $x^{2} + bx + c$, look for two numbers that multiply to $c$ (which is $20$) and add up to $b$ (which is $-9$): $-5$ and $-4$.
Step:1.2.1.2
Express the factored form with these numbers: $(x - 5)(x - 4) = 0$.
Step:1.2.2
The equation is satisfied if any factor equals zero: $x - 5 = 0$ or $x - 4 = 0$.
Step:1.2.3
Isolate $x$ when $x - 5 = 0$.
Step:1.2.3.1
Set $x - 5$ to zero: $x - 5 = 0$.
Step:1.2.3.2
Solve for $x$ by adding $5$: $x = 5$.
Step:1.2.4
Isolate $x$ when $x - 4 = 0$.
Step:1.2.4.1
Set $x - 4$ to zero: $x - 4 = 0$.
Step:1.2.4.2
Solve for $x$ by adding $4$: $x = 4$.
Step:1.2.5
The values that satisfy $(x - 5)(x - 4) = 0$ are $x = 5$ and $x = 4$.
Step:1.3
The domain excludes the values that make the denominator zero. Thus, the domain in interval notation is $(-\infty, 4) \cup (4, 5) \cup (5, \infty)$, and in set-builder notation is $\{x | x \neq 4, x \neq 5\}$.
Step:2
The function $\frac{x - 5}{x^{2} - 9x + 20}$ is not continuous for all real numbers due to the domain restrictions.
Step:3
Knowledge Notes:
To determine if a function is continuous, one must first establish its domain, which is the set of all input values for which the function is defined. For rational functions, like the one given, continuity issues often arise where the denominator equals zero, as division by zero is undefined.
The process of finding the domain typically involves setting the denominator equal to zero and solving for the variable. If the denominator can be factored, as in this case, the roots of the denominator indicate the values that are not in the domain.
In this problem, the denominator is a quadratic expression that can be factored using the AC method, which is a technique for factoring quadratics where the leading coefficient is 1. The method involves finding two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b).
Once the roots are found, they are excluded from the domain, and the domain is expressed in interval notation (which specifies the range of values in intervals) or set-builder notation (which defines a set by a property that its members must satisfy).
A function is continuous at a point if the limit of the function as it approaches the point from both the left and the right exists and equals the function's value at that point. If the domain of the function has any restrictions, such as excluded values where the function is undefined, it cannot be continuous over the entire set of real numbers.
