Determine if Continuous f(x)=8/(x-8)-9x
The problem provided asks to analyze the continuous function f(x) = 8/(x-8) - 9x and determine whether it is continuous for all values of x. This involves checking if the function is defined and smooth (without breaks, holes, or jumps) at every point on its domain. The question specifically focuses on mathematical continuity, which requires that for every point 'c' in the domain, the limit of f(x) as x approaches 'c' equals f(c). The function might exhibit discontinuities at certain points if, for example, there are values of x that cause division by zero or if there are other conditions that cause the function not to be well-defined.
$f \left(\right. x \left.\right) = \frac{8}{x - 8} - 9 x$
Answer
Solution:
Step:1
Identify the domain to check the continuity of the function.
Step:1.1
To find the discontinuities, solve for $x$ when the denominator of $\frac{8}{x - 8}$ is zero. $x - 8 = 0$
Step:1.2
Isolate $x$ by adding $8$ to both sides. $x = 8$
Step:1.3
The function is defined for all $x$ except where it is undefined. In Interval Notation: $(-\infty, 8) \cup (8, \infty)$ In Set-Builder Notation: $\{x | x \neq 8\}$
Step:2
The function $\frac{8}{x - 8} - 9x$ does not have a domain of all real numbers, indicating it is not continuous everywhere.
Step:3
Knowledge Notes:
To determine the continuity of a function, 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 in the problem $\frac{8}{x - 8} - 9x$, discontinuities occur where the denominator is zero, as division by zero is undefined.
In the given function, the denominator is $x - 8$. Setting this equal to zero and solving for $x$ gives the value at which the function is not defined, which in this case is $x = 8$. This means the function is not continuous at $x = 8$.
The domain of the function is expressed using interval notation and set-builder notation. Interval notation represents the domain as a union of intervals, excluding the point of discontinuity. Set-builder notation expresses the domain as a set of values that satisfy a given condition, in this case, $x \neq 8$.
A function is continuous if it is defined and uninterrupted at all points within its domain. Since the function in question is not defined at $x = 8$, it is not continuous over the entire set of real numbers.
