Determine if Continuous f(x)=8/(x^2-2x-35)
The given problem is to assess whether the function f(x) = 8/(x^2 - 2x - 35) is continuous. Essentially, this involves checking if the function does not have any breaks, holes, or jumps at any point within its domain. In the context of the function provided, one would pay special attention to the points where the denominator might become zero (since division by zero is undefined in mathematics), which can lead to discontinuities. The task would involve analyzing the behavior of the function around these critical points and throughout the rest of its domain.
$f \left(\right. x \left.\right) = \frac{8}{x^{2} - 2 x - 35}$
Answer
Solution:
Determine the Continuity of the Function $f(x)=\frac{8}{x^2-2x-35}$
Step 1: Identify the Domain for Continuity
1.1. To find the domain, set the denominator of $\frac{8}{x^2 - 2x - 35}$ to zero: $x^2 - 2x - 35 = 0$.
1.2. Proceed to solve for $x$.
1.2.1. Apply the AC factoring method to $x^2 - 2x - 35$.
1.2.1.1. Look for two numbers that multiply to $c$ and add up to $b$, where $c = -35$ and $b = -2$. The numbers are $-7$ and $5$.
1.2.1.2. Use these numbers to express the quadratic in its factored form: $(x - 7)(x + 5) = 0$.
1.2.2. The equation equals zero when any factor equals zero: $x - 7 = 0$ or $x + 5 = 0$.
1.2.3. Solve for $x$ when $x - 7 = 0$.
1.2.3.1. Isolate $x$: $x - 7 = 0$.
1.2.3.2. Add $7$ to both sides to find $x$: $x = 7$.
1.2.4. Solve for $x$ when $x + 5 = 0$.
1.2.4.1. Isolate $x$: $x + 5 = 0$.
1.2.4.2. Subtract $5$ from both sides to find $x$: $x = -5$.
1.2.5. The roots of the equation $(x - 7)(x + 5) = 0$ are $x = 7$ and $x = -5$.
1.3. The domain excludes the values where the denominator is zero. Thus, the domain in interval notation is $(-\infty, -5) \cup (-5, 7) \cup (7, \infty)$, and in set-builder notation is $\{x | x \neq -5, 7\}$.
Step 2: Assess Continuity Over the Domain
Since the domain excludes $x = -5$ and $x = 7$, the function $\frac{8}{x^2 - 2x - 35}$ is not continuous over the entire set of real numbers.
Step 3: Conclusion
The function has discontinuities at $x = -5$ and $x = 7$, and therefore is not continuous for all real numbers.
Knowledge Notes:
To determine the continuity of a rational function like $f(x) = \frac{8}{x^2 - 2x - 35}$, one must first establish its domain, which consists of all real numbers except those that make the denominator equal to zero. The steps involve:
Setting the denominator equal to zero and solving the resulting quadratic equation to find the values that are not in the domain.
Factoring the quadratic equation, if possible, to find its roots more easily.
Using interval notation to express the domain, which excludes the roots found in step 1.
Recognizing that if the domain of the function does not include all real numbers, then the function is not continuous over the entire real number line.
The AC method mentioned in the solution is a factoring technique used to factor quadratic equations of the form $ax^2 + bx + c$. It involves finding two numbers that multiply to $a \times c$ and add up to $b$. These two numbers are then used to split the middle term and factor by grouping.
Interval notation is a way of writing subsets of the real number line. An interval that does not include its endpoints is written with parentheses, while an interval that includes its endpoints is written with square brackets.
Set-builder notation is another way to describe a set, defining the properties that its members must satisfy. For example, $\{x | x \neq -5, 7\}$ describes the set of all $x$ such that $x$ is not equal to $-5$ or $7$.
