Question

What points, if any, are the functions discontinuous? $$ f(u)=\frac{2 u+7}{\sqrt{u+5}} $$

Short answer

The function is discontinuous at \( u = -5 \).

Step-by-step solution

Identify discontinuity types

A function can be discontinuous where it's not defined. For rational functions like \( f(u) = \frac{2u + 7}{\sqrt{u + 5}} \), discontinuities occur when the denominator is zero or not defined. Recognize that the denominator is \( \sqrt{u + 5} \), which is undefined for negative numbers under the square root.

Find the domain of the denominator

The square root function \( \sqrt{u + 5} \) is defined for \( u + 5 \geq 0 \), which simplifies to \( u \geq -5 \). This means the function \( f(u) \) is defined only for \( u \geq -5 \).

Check for zero in the denominator

At the point where \( u = -5 \), the expression \( \sqrt{u + 5} = \sqrt{0} = 0\). Since you cannot divide by zero, there is a discontinuity at \( u = -5 \).

Check within the simplified domain

Within the domain \( u \geq -5 \), any other point does not cause the denominator to be zero. Thus, \( f(u) \) is continuous for \( u > -5 \).

Key concepts

Rational Functions

A rational function is a type of function that is represented as the quotient of two polynomials. It takes the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not the zero polynomial. The presence of a polynomial in the denominator introduces potential points of discontinuity, primarily where \( Q(x) = 0 \). This is because division by zero is undefined in mathematics.

Key characteristics of rational functions include:

• They may have vertical asymptotes, where the function tends to infinity as it approaches a certain value.
• They can also have holes, which are points of discontinuity where both the numerator and the denominator simultaneously become zero.

For the function \( f(u) = \frac{2u + 7}{\sqrt{u + 5}} \), it's crucial to examine the denominator for these aspects, given the square root function it contains.

Domain of a Function

The domain of a function is the set of all possible input values (typically \( x \) values) for which the function is defined. Determining the domain involves examining constraints such as division by zero, square roots of negative numbers, and other limitations inherent in the function.

For the function \( f(u) = \frac{2u + 7}{\sqrt{u + 5}} \), the domain is restricted by the square root in the denominator. The square root function \( \sqrt{u + 5} \) is only defined for non-negative values under the root, leading to the inequality \( u + 5 \geq 0 \) or \( u \geq -5 \).

This means the domain of \( f(u) \) is all \( u \) values that are greater than or equal to \(-5\). Any value lesser than \(-5\) would lead to a negative number under the square root, making the denominator undefined.

Square Root Function

The square root function is a mathematical function that returns the non-negative root of a number. It is usually represented by \( \sqrt{x} \), which denotes the principal (or positive) square root. It has a few important properties concerning its definition and domain:

- The square root function is defined only for non-negative arguments. Thus, the condition \( x \geq 0 \) must be satisfied for \( \sqrt{x} \) to have a real number output.
- It is considered a monotonically increasing function on its domain, meaning as the input value increases, the output value also increases.

In the given function \( f(u) = \frac{2u + 7}{\sqrt{u + 5}} \), the square root \( \sqrt{u + 5} \) must have non-negative inputs. Therefore, \( u + 5 \geq 0 \) or \( u \geq -5 \) constrains the domain. Additionally, at \( u = -5 \), the square root equals zero, which causes division by zero, highlighting a critical point of discontinuity.

Continuous Functions

Continuous functions are functions that do not have interruptions in their graphs. That is, you can draw the graph of a continuous function without lifting your pencil from the paper. This property ensures that small changes in the input produce small changes in the output. In mathematical terms, a function \( f(x) \) is continuous at a point \( a \) if the limit \( \lim_{x \to a} f(x) = f(a) \).

However, not all functions are continuous everywhere. Discontinuities in functions may arise at points where:

• The function is not defined.
• The limit does not exist.
• The limit exists but does not equal the function's value at that point.

Looking at \( f(u) = \frac{2u + 7}{\sqrt{u + 5}} \), it is continuous for \( u > -5 \) since the denominator never becomes zero. The problem arises specifically at \( u = -5 \), where division by zero occurs, indicating a discontinuity. For values less than \(-5\), the function is altogether undefined due to the negative value inside the square root. Hence, it's vital to understand these interruptions when analyzing function continuity.