Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. \(f(x)=\frac{x+1}{\sqrt{x}}\)

Step-by-Step Solution

Answer

The function \(f(x)=\frac{x+1}{\sqrt{x}}\) is continuous for \(x\) in the interval \((0, +∞)\).

Step 1: Function Analysis

The function given is \(f(x)=\frac{x+1}{\sqrt{x}}\). The denominator here is \(\sqrt{x}\). Hence, to understand the continuity of the function, first find for what values of \(x\) the denominator becomes zero.

Step 2: Identify Discontinuity Point

Set the denominator equal to zero and solve for \(x\). \(\sqrt{x}=0\), squaring both sides we get \(x=0\). Therefore, \(x=0\) is a point of discontinuity.

Step 3: Identify Continuity Intervals

Therefore, the function is continuous for all real values of \(x\) except \(x=0\). However, since we have a square root in the denominator, we should also keep in mind that \(x\) must be greater than or equal to zero, as square root of a negative number is undefined. Therefore, the continuous intervals are \(x\) in \((0, +∞)\).

Key Concepts

Functions in Calculus

Functions are the building blocks of calculus and help us understand how quantities change and relate to each other. In calculus, a function is a rule that assigns each input exactly one output. Some common types of functions you will encounter include linear, polynomial, exponential, logarithmic, and trigonometric functions. It is important to understand whether a function is continuous or not, as this affects how it can be used in calculus.

A continuous function is one where small changes in the input result in small changes in the output, without any interruptions or jumps. This is important because continuity ensures that a graph of the function can be drawn without lifting your pencil. Mathematically, a function \(f(x)\) is continuous at a point \(x=a\) if the following conditions are satisfied:

• The function \(f(x)\) is defined at \(x=a\).
• The limit of \(f(x)\) as \(x\) approaches \(a\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(a\) equals \(f(a)\).

Understanding continuity helps us analyze functions and predict their behavior over intervals.

Discontinuity

Discontinuity occurs when a function is not continuous at some point in its domain. There are different types of discontinuities that you might encounter:

• Removable Discontinuity: Occurs if you can "fix" the discontinuity by redefining the function at a certain point. This often appears as a hole in the graph.
• Jump Discontinuity: This is when there is a sudden change in function values, causing a break in the graph.
• Infinite Discontinuity: Happens when the function approaches infinity at a point, usually resulting in a vertical asymptote.

In the function \(f(x)=\frac{x+1}{\sqrt{x}}\), the point of discontinuity occurs at \(x=0\) because at this point, the function is undefined due to the square root in the denominator, leading to division by zero. Hence, \(x=0\) represents an infinite discontinuity. Identifying discontinuities is crucial for understanding where a function may behave erratically or unpredictably.

Square Root Function

The square root function, represented as \(\sqrt{x}\), is a function that assigns to each non-negative number \(x\) its principal square root. The square root function is defined only for non-negative values of \(x\), since the square root of a negative number is not defined in the set of real numbers.

In the context of continuity, the presence of a square root, such as in \(f(x)=\frac{x+1}{\sqrt{x}}\), imposes restrictions on the range of \(x\) values where the function is defined. Since \(\sqrt{x}\) is part of the denominator, it also affects the continuity of the entire function. This means \(x\) must be greater than zero for \(f(x)\) to be real and defined. Therefore, the function is continuous only for \(x\) in the interval \((0, \infty)\).

Understanding the properties of the square root function, such as its domain and how it interacts with other functions in a formula, helps in determining where that function can be considered continuous, and identifying any discontinuities it might have.