Determine if Continuous f(x)=(x^2-7x+12)/(x^2-16)
The problem here asks for the determination of whether the function f(x) = (x^2 - 7x + 12) / (x^2 - 16) is continuous. This involves checking if the function f(x) does not have any breaks, jumps, or holes throughout its domain. The domain of the function is all real numbers for which the function is defined. Continuity typically requires checking for points where the function could potentially be undefined (in this case, where the denominator is zero), and seeing if the function approaches the same value from both the left and the right at these points.
$f \left(\right. x \left.\right) = \frac{x^{2} - 7 x + 12}{x^{2} - 16}$
Answer
Solution:
Step:1
Identify the domain to check for continuity of the function.
Step:1.1
To find the discontinuities, equate the denominator of $\frac{x^{2} - 7x + 12}{x^{2} - 16}$ to zero: $x^{2} - 16 = 0$.
Step:1.2
Determine the values of $x$ that satisfy the equation.
Step:1.2.1
Add $16$ to both sides to isolate $x^{2}$: $x^{2} = 16$.
Step:1.2.2
Extract the square root of both sides to solve for $x$: $x = \pm \sqrt{16}$.
Step:1.2.3
Simplify the square root.
Step:1.2.3.1
Express $16$ as a square of an integer: $x = \pm \sqrt{4^{2}}$.
Step:1.2.3.2
Remove the square root by taking out the squared term: $x = \pm 4$.
Step:1.2.4
Combine both the positive and negative solutions for the complete set.
Step:1.2.4.1
Apply the positive part of $\pm$ to find the first solution: $x = 4$.
Step:1.2.4.2
Apply the negative part of $\pm$ to find the second solution: $x = -4$.
Step:1.2.4.3
Combine both solutions: $x = 4, -4$.
Step:1.3
The domain includes all $x$ values that keep the function defined.
Interval Notation: $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$ Set-Builder Notation: $\{x | x \neq 4, -4\}$
Step:2
As the domain excludes some real numbers, the function $\frac{x^{2} - 7x + 12}{x^{2} - 16}$ is not continuous for all real numbers.
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. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. If there's a point where the function is not defined or the limit does not exist or does not equal the function's value, the function is not continuous at that point.
For rational functions, such as the one given in the problem, discontinuities arise where the denominator is zero, as division by zero is undefined. Thus, finding the values that make the denominator zero is a crucial step in determining the domain.
The domain can be expressed using interval notation, which describes the set of numbers between two endpoints, and set-builder notation, which specifies a set by a property that its members must satisfy.
The function given in the problem, $\frac{x^{2} - 7x + 12}{x^{2} - 16}$, is a rational function, and its continuity depends on the values that do not cause division by zero. The denominator $x^{2} - 16$ is zero when $x = \pm 4$, so the function is not continuous at these points. The domain of the function is all real numbers except $x = 4$ and $x = -4$. Since there are values excluded from the domain, the function is not continuous over the entire set of real numbers.
