Determine the interval(s) on which the following functions are continuous. 
f(x)=x^5+6x+17 / x^2−9

Verified step by step guidance

1. Identify the type of function: \( f(x) = \frac{x^5 + 6x + 17}{x^2 - 9} \) is a rational function, which is continuous everywhere except where the denominator is zero.
2. Set the denominator equal to zero to find the points of discontinuity: \( x^2 - 9 = 0 \).
3. Solve the equation \( x^2 - 9 = 0 \) to find the values of \( x \) that make the denominator zero. This can be factored as \( (x - 3)(x + 3) = 0 \).
4. Determine the solutions to the equation: \( x = 3 \) and \( x = -3 \). These are the points where the function is not continuous.
5. Conclude that the function \( f(x) \) is continuous on the intervals \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

Key Concepts

Continuity of Functions

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. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The continuity of rational functions is affected by the values that make the denominator zero, as these points create vertical asymptotes or discontinuities. Understanding the behavior of rational functions helps in identifying intervals of continuity.

Finding Intervals of Continuity

To find the intervals of continuity for a function, one must identify points where the function is undefined, typically where the denominator is zero. After determining these points, the intervals can be established by testing the function's behavior in the regions between these points. This process allows for a comprehensive understanding of where the function remains continuous.