Analysis Of The Function F(x) = √x / (x + 1)

This article delves into a comprehensive analysis of the function $f(x) = \frac{\sqrt{x}}{x+1}$. We will explore various aspects of this function, including evaluating it at specific points, determining its domain, and calculating its average rate of change. This exploration will provide a thorough understanding of the function's behavior and characteristics. This article aims to provide a clear and concise explanation of each step, making it accessible to anyone interested in understanding the intricacies of this function.

(a) Evaluating f(0), f(3), and f(a+3)

Understanding Function Evaluation

To begin our analysis, we will evaluate the function $f(x)$ at specific points: $x = 0$, $x = 3$, and $x = a + 3$. Function evaluation is a fundamental concept in mathematics, where we substitute a given value for the variable $x$ in the function's expression and simplify to find the corresponding output. This process allows us to understand how the function behaves at different input values and provides valuable insights into its overall behavior. The function $f(x)$ involves both a square root and a rational expression, so careful consideration must be given to the domain and potential restrictions. The domain of a function is the set of all possible input values ($x$-values) for which the function is defined. In this case, the square root requires the argument to be non-negative, and the denominator cannot be zero. Therefore, we need to ensure that $x \geq 0$ and $x + 1 \neq 0$. These conditions will guide our evaluation process and help us determine the validity of our results. Evaluating functions at specific points is a crucial step in understanding their properties and applications in various mathematical and real-world contexts. It forms the basis for further analysis, such as graphing, finding rates of change, and solving equations involving the function. In the subsequent sections, we will perform these evaluations step-by-step, providing clear explanations and justifications for each step.

Evaluating f(0)

To evaluate $f(0)$, we substitute $x = 0$ into the function:

$f(0) = \frac{\sqrt{0}}{0 + 1} = \frac{0}{1} = 0$

Therefore, $f(0) = 0$. This indicates that the function has a value of 0 when $x$ is 0. This point lies on the graph of the function and provides a starting point for understanding its behavior. The simplicity of this evaluation highlights the importance of understanding the function's expression and the properties of basic mathematical operations. Substituting 0 into the square root yields 0, and the denominator becomes 1, resulting in a final value of 0. This result is consistent with the domain of the function, as 0 is a valid input value. The evaluation of $f(0)$ serves as a foundation for further analysis, as it provides a known point on the function's graph and helps in visualizing its overall shape and behavior. Understanding the value of a function at specific points is crucial for various applications, such as finding intercepts, determining intervals of increase and decrease, and solving optimization problems. In this case, the fact that $f(0) = 0$ indicates that the function passes through the origin, which is a significant characteristic for many functions.

Evaluating f(3)

To evaluate $f(3)$, we substitute $x = 3$ into the function:

$f(3) = \frac{\sqrt{3}}{3 + 1} = \frac{\sqrt{3}}{4}$

Therefore, $f(3) = \frac{\sqrt{3}}{4}$. This value represents the function's output when the input is 3. It's an irrational number, as it involves the square root of 3. This evaluation demonstrates how the function handles non-integer inputs and provides another point on the function's graph. The square root of 3 is a positive number, and adding 1 to 3 in the denominator results in 4. The fraction $\frac{\sqrt{3}}{4}$ is a precise representation of the function's value at $x = 3$, and it can be approximated as a decimal if needed. This evaluation further expands our understanding of the function's behavior and its range of possible outputs. Knowing the value of the function at $x = 3$ helps in sketching its graph and understanding its rate of change. In many applications, such as modeling physical phenomena, the value of a function at specific points can represent crucial information about the system being modeled. The evaluation of $f(3)$ illustrates the function's ability to handle both integer and non-integer inputs and produce corresponding outputs that reflect its underlying mathematical structure.

Evaluating f(a+3)

To evaluate $f(a+3)$, we substitute $x = a + 3$ into the function:

$f(a+3) = \frac{\sqrt{a+3}}{(a+3) + 1} = \frac{\sqrt{a+3}}{a+4}$

Therefore, $f(a+3) = \frac{\sqrt{a+3}}{a+4}$. This expression represents the function's value when the input is $a + 3$. It introduces a variable $a$ into the function's argument, making the output an expression in terms of $a$. This type of evaluation is crucial for understanding how the function behaves with variable inputs and for analyzing its general properties. The expression $\sqrt{a+3}$ in the numerator requires $a + 3$ to be non-negative, meaning $a \geq -3$. The denominator $a + 4$ cannot be zero, so $a \neq -4$. These conditions define the valid range of values for $a$ in this context. The evaluation of $f(a+3)$ allows us to express the function's output as a function of $a$, which can be used for further analysis, such as finding limits, derivatives, and integrals. Understanding how a function transforms variable inputs is essential for many mathematical and scientific applications. In this case, the evaluation of $f(a+3)$ provides a general expression that captures the function's behavior for a range of input values determined by the variable $a$. This generalized evaluation is a powerful tool for exploring the function's properties and its relationship to other mathematical concepts.

(b) Finding the Domain of f

Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function $f(x) = \frac{\sqrt{x}}{x+1}$, we need to consider two restrictions:

1. The expression inside the square root must be non-negative: $x \geq 0$.
2. The denominator cannot be zero: $x + 1 \neq 0$, which means $x \neq -1$.

Combining these restrictions, we have $x \geq 0$ and $x \neq -1$. Since $x$ must be greater than or equal to 0, the condition $x \neq -1$ is already satisfied. Therefore, the domain of $f$ is all non-negative real numbers. The domain of a function is a fundamental concept in mathematics, as it defines the set of values for which the function produces valid outputs. Understanding the domain is crucial for interpreting the function's behavior and applying it in various contexts. In this case, the square root function and the rational expression impose restrictions on the possible input values. The square root function, denoted by $\sqrt{x}$, is only defined for non-negative real numbers, as the square root of a negative number is not a real number. This restriction leads to the condition $x \geq 0$. The rational expression, represented by the fraction $\frac{\sqrt{x}}{x+1}$, has a denominator of $x + 1$. A fraction is undefined when its denominator is equal to zero, so we must ensure that $x + 1 \neq 0$. Solving this inequality gives us $x \neq -1$. These two restrictions, $x \geq 0$ and $x \neq -1$, define the domain of the function. Combining these restrictions is essential to determine the overall domain. Since $x$ must be greater than or equal to 0, the condition $x \neq -1$ is automatically satisfied, as -1 is not within the interval of non-negative real numbers. Therefore, the domain of $f(x)$ is the set of all non-negative real numbers.

Expressing the Domain in Interval Notation

The domain of $f$ in interval notation is $[0, \infty)$. Interval notation is a concise way to represent sets of real numbers using intervals. A closed bracket, such as [, indicates that the endpoint is included in the interval, while an open parenthesis, such as (, indicates that the endpoint is not included. The symbol $\infty$ represents infinity, indicating that the interval extends indefinitely in the positive direction. In this case, the interval $[0, \infty)$ represents all real numbers greater than or equal to 0. The left endpoint, 0, is included because the function is defined at $x = 0$. The right endpoint, $\infty$, is not included, as infinity is not a real number but rather a concept representing unbounded growth. Using interval notation allows us to express the domain of the function in a clear and unambiguous manner. The interval $[0, \infty)$ captures the essential characteristic of the domain, which is the set of all non-negative real numbers. Understanding interval notation is crucial for working with mathematical functions and expressing their properties concisely. In various applications, such as calculus and analysis, interval notation is frequently used to describe intervals of interest, such as domains, ranges, and intervals of convergence. The use of brackets and parentheses provides a precise way to indicate whether endpoints are included or excluded, which is essential for accurate mathematical communication. Therefore, expressing the domain of $f$ as $[0, \infty)$ provides a complete and concise representation of the set of all possible input values for the function.

(c) Average Rate of Change

Understanding Average Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula:

$\frac{f(b) - f(a)}{b - a}$

This formula represents the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ on the graph of the function. It measures the average change in the function's output with respect to a change in its input over the specified interval. The average rate of change is a fundamental concept in calculus and is used to approximate the instantaneous rate of change, which is the derivative. The average rate of change provides valuable information about how the function's output changes on average over a given interval. It is a measure of the function's steepness or slope between two points. The formula for the average rate of change involves calculating the difference in the function's values at the endpoints of the interval, $f(b) - f(a)$, and dividing it by the difference in the corresponding input values, $b - a$. This calculation is analogous to finding the slope of a line, which is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). The average rate of change can be positive, negative, or zero, depending on the function's behavior over the interval. A positive average rate of change indicates that the function is increasing on average, while a negative average rate of change indicates that the function is decreasing on average. An average rate of change of zero means that the function's output does not change on average over the interval. Understanding the average rate of change is crucial for various applications, such as modeling physical phenomena, analyzing data, and solving optimization problems. It provides a way to quantify how a function's output changes in response to changes in its input, which is essential for making predictions and understanding trends.

Calculating the Average Rate of Change

To calculate the average rate of change, we need to identify the interval $[a, b]$. Let's consider a general interval $[a, a+h]$, where $h$ represents a change in $x$. The average rate of change over this interval is:

$\frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}$

This expression gives us a general formula for the average rate of change of $f(x)$ over the interval $[a, a+h]$. By substituting specific values for $a$ and $h$, we can calculate the average rate of change over different intervals. The interval $[a, a+h]$ represents a generic interval where $a$ is the starting point and $h$ is the width of the interval. This notation allows us to express the average rate of change in a general form, which can then be applied to specific intervals by substituting the appropriate values for $a$ and $h$. The expression $f(a+h)$ represents the function's value at the endpoint $a+h$, while $f(a)$ represents the function's value at the starting point $a$. The difference $f(a+h) - f(a)$ represents the change in the function's output over the interval. The denominator, $h$, represents the change in the input variable $x$ over the interval. Dividing the change in the output by the change in the input gives us the average rate of change. This formula provides a versatile tool for analyzing the function's behavior over different intervals. By choosing specific values for $a$ and $h$, we can calculate the average rate of change over intervals of varying lengths and positions. For example, we can examine the average rate of change over small intervals to approximate the instantaneous rate of change or over larger intervals to understand the function's overall trend. The general formula for the average rate of change is a fundamental concept in calculus and is used extensively in various applications, such as finding derivatives, analyzing function behavior, and solving optimization problems.

Applying the Formula to Our Function

Substituting $f(x) = \frac{\sqrt{x}}{x+1}$ into the average rate of change formula, we get:

$\frac{\frac{\sqrt{a+h}}{(a+h)+1} - \frac{\sqrt{a}}{a+1}}{h} = \frac{\frac{\sqrt{a+h}}{a+h+1} - \frac{\sqrt{a}}{a+1}}{h}$

This expression represents the average rate of change of the function $f(x)$ over the interval $[a, a+h]$. Simplifying this expression further can provide insights into the function's behavior. The expression $\frac{\sqrt{a+h}}{a+h+1}$ represents the function's value at the endpoint $a+h$, while $\frac{\sqrt{a}}{a+1}$ represents the function's value at the starting point $a$. The difference between these two values, divided by $h$, gives us the average rate of change. Simplifying this complex fraction involves finding a common denominator for the two fractions in the numerator and then dividing by $h$. This process can be algebraically intensive but can lead to a more compact and insightful expression for the average rate of change. The simplified expression can reveal how the average rate of change depends on the values of $a$ and $h$. For example, it can show how the average rate of change changes as the interval width $h$ approaches zero, which is related to the concept of the derivative. Understanding the average rate of change of a function is crucial for various applications, such as analyzing function behavior, making approximations, and solving optimization problems. It provides a way to quantify how the function's output changes in response to changes in its input, which is essential for modeling real-world phenomena.

This detailed analysis provides a thorough understanding of the function $f(x) = \frac{\sqrt{x}}{x+1}$, including its evaluation at specific points, its domain, and its average rate of change. Each aspect contributes to a comprehensive view of the function's behavior and characteristics.