Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x-5}{x^{2}-9 x+20}\)

Short answer

The function is continuous on the intervals \(-\infty, 4\) and \(4, \infty\). It has an infinite discontinuity at \(x=4\).

Step-by-step solution

Simplify the Function

Start by simplifying the given function. The denominator can be factored as \(x^{2}-9x+20 = (x-4)(x-5)\). So, the function can be rewritten as \(f(x)=\frac{x-5}{(x-4)(x-5)}\). Simplifying further removes the (x-5) term from the numerator and denominator, rendering the function as \(f(x)=\frac{1}{x-4}\).

Identify Points of Discontinuity

A function is not defined where the denominator equals zero, which are points of discontinuity. Since the denominator here is \(x-4\), the function is not defined at \(x=4\). This point breaks the domain of the function into two intervals, \(-\infty, 4\) and \(4, \infty\).

Establish the Continuity of the Function

Rational functions are continuous everywhere except where the denominator equals zero. Therefore, this function is continuous on the intervals \(-\infty, 4\) and \(4, \infty\). Because the function is not defined at \(x=4\), the function is discontinuous at that point, which is a non-removable discontinuity or infinite discontinuity.

Key concepts

Discontinuity

Understanding where a function is not smooth or interrupted is as vital as knowing where it is continuous. Discontinuity occurs whenever a function fails to be continuous at a point within its domain. In mathematical terms, a function is considered discontinuous at a point when it does not satisfy the three conditions of continuity:

• The function must be defined at the point. If there’s no value assigned to the function at that point, the function can’t be continuous there.
• The limit of the function as it approaches the point must exist. If the function behaves erratically as it nears the point, it can’t be continuous.
• The function’s value at the point must equal the limit as it approaches the point. Even if both the value and the limit exist separately, they must also match for the function to be continuous.

Whenever a function violates one or more of these conditions, we have identified a discontinuity, which can come in various forms such as jump, infinite, or removable discontinuities.

Rational Functions

Rational functions are algebraic expressions defined by the ratio of two polynomials, where the numerator and denominator are both polynomials and the denominator is not zero. The general form is expressed as \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). What makes rational functions especially interesting is their behavior near values that make the denominator zero, as these points are not included in the function's domain and can cause discontinuities. To study these functions closely, one must analyze their limits, intercepts, asymptotic behavior, and intervals of continuity to fully understand their overall behavior.

Limits in Calculus

Limits are foundational in calculus for analyzing how a function behaves near a certain point, and they play a critical role in understanding discontinuities. A limit is an estimated value that a function approaches as the input approaches some value. For instance, the limit of \( f(x) \) as x approaches c is the value that \( f(x) \) gets closer to as x gets infinitely close to c. Properly evaluating limits allows us to deal with situations where direct substitution is not possible due to points of discontinuity. Calculating limits helps identify if a discontinuity is removable (where a limit exists but the function is not defined) or non-removable (where the limit does not exist, like at a vertical asymptote).

Points of Discontinuity

In context with rational functions, points of discontinuity are specific values in the domain of a function at which the function ceases to be continuous. These are often the values that would make the denominator of the rational function zero, as in the case of \( f(x)=\frac{1}{x-4} \) where the point of discontinuity is \( x=4 \) since the function is undefined there. Identifying these points is crucial because they outline where the function behaves irregularly. The behavior near these points is often studied using limits, which can tell us if a discontinuity is a 'hole' in the function, a vertical intercept or asymptote, or a jump from one value to another. Graphically, these are the x-values where the function graph is interrupted or has an asymptote.