Question

Determining Differentiability In Exercises \(75-80\) , describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\frac{2}{x-3} $$

Short answer

The function \(f(x) = \frac{2}{x-3}\) is differentiable for all real numbers except at \(x = 3\).

Step-by-step solution

Identifying Discontinuity

First, let's find where there might be a discontinuity in the function. This occurs when the denominator of the function is equal to zero. In this case we can write out that \(x-3 = 0\). Solving this, we find \(x = 3\). Therefore, the function has a discontinuity at \(x = 3\).

Checking Differentiability

Differentiability exists everywhere in the domain of the function excluding the points of discontinuity. As we determined that the function has a discontinuity at \(x = 3\), this means the function is not differentiable at \(x = 3\).

Describing Values

Therefore, the function \(f(x) = \frac{2}{x-3}\) is differentiable for all real numbers except for \(x = 3\).

Key concepts

Discontinuity

In the realm of mathematics, especially while dealing with functions, the concept of discontinuity is essential. Discontinuity in a function occurs when there is an abrupt change in the behavior or output of the function at a particular point. To put it simply, a function is said to be discontinuous at a point if the function does not have a defined value or if the value jumps at that point.

To identify discontinuity in a function, one approach is to check where the denominator equals zero for rational functions. For example, in the function \(f(x) = \frac{2}{x-3}\), setting the denominator zero, \(x-3 = 0\), reveals a discontinuity at \(x = 3\).

Discontinuities can come in several forms, such as:

• Jump Discontinuity: When there is a sudden jump in the function values.
• Infinite Discontinuity: Usually seen in rational functions where the function values tend toward infinity as x approaches a point.
• Removable Discontinuity: When a point can be added or adjusted to make the function continuous.

Recognizing these forms helps understand whether the function behaves predictably around a particular point.

Differentiable Functions

The notion of differentiability is integral in calculus. A function is considered differentiable at a point if it has a defined derivative there. This implies the function's graph is smooth and has no sharp corners or cusps at that point.

Differentiability implies continuity, but the converse isn't always true. For example, while a continuous function might be "smooth" over a section of its domain, it may not have a derivative at some points due to discontinuity.

In the case of the function \(f(x) = \frac{2}{x-3}\), its differentiability depends on its continuity. We determined that the function is discontinuous at \(x = 3\); therefore, it's not differentiable there. The lack of a derivative at \(x = 3\) highlights a break in the graph's slope at this point, illustrating where the function does not possess a tangent.

In general, a function is differentiable:

• Everywhere it's continuous.
• Where there are no sharp edges or vertical tangents.

Grasping differentiability helps in understanding the changes in a function, forming a basis for many calculus applications.

Domain of a Function

Understanding the domain of a function is foundational in learning about functions. The domain defines all the input values (x-values) for which the function is defined. It essentially answers the question: For which x-values does the function work without any issues?

For our function \(f(x) = \frac{2}{x-3}\), the domain consists of all real numbers except where the denominator, \(x - 3\), equals zero. This exclusion arises because division by zero is undefined in mathematics.

Thus, the domain of this function is all real numbers except \(x = 3\) due to the discontinuity. In mathematical notation, we write this as \(\{x \in \mathbb{R} \mid x eq 3\}\).

Paying attention to the domain is crucial for properly defining and manipulating functions. It ensures clarity about where a function operates normally and where it might face issues, such as undefined operations or discontinuities.