Problem 61 - Algebra for College Students

Question

\(f(x)=\sqrt{x^{2}+1}-4\)

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt{x^2 + 1} - 4\) is defined for all real numbers with a minimum value of \(-3\).

Step 1: Understand the Problem

We need to find a solution or determine an expression based on the given function \(f(x) = \sqrt{x^{2} + 1} - 4\). Let's consider this function and understand its components: a square root expression and a constant.

Step 2: Analyze the Function

We can see the function consists of two parts: the square root \(\sqrt{x^2 + 1}\) and the constant \(-4\). This means we are actually looking at a vertical shift of the square root function.

Step 3: Identify Domain and Range

Since the square root function is defined for all real numbers under its square root portion \(x^2 + 1\), \(x\) can be any real number. The minimum value of \(f(x)\) occurs when the square root is at its minimum, which is \(\sqrt{1} = 1\). Therefore, the minimum value of \(f(x)\) is \(1 - 4 = -3\).

Step 4: Simplify or Transform (if needed)

In this case, we don't need further simplification. However, we can understand that the function is already simplified. The expression itself is simply a transformation of the root function.

Step 5: Graph & Interpretation (Optional)

If we were to graph this function, it would resemble a slightly transformed version of a square root function, started at \(-4\) instead of \(0\) due to the vertical shift down by 4. The lowest the function goes is \(-3\) due to its nature.

Key Concepts

Square Root Function

The square root function is a fundamental concept in algebra, represented by the symbol \( \sqrt{} \). It is defined mathematically as the function \( f(x) = \sqrt{x} \). The square root function aims to find a number which, when multiplied by itself, results in the given number. For instance, the square root of 9 is 3, because \( 3^2 = 9 \). This function is crucial because it allows us to understand and solve equations involving squares.

In the context of our particular function, \( f(x) = \sqrt{x^2 + 1} - 4 \), the square root \( \sqrt{x^2 + 1} \) involves a sum within the square root. The sum ensures that the expression inside the square root is always positive or zero, which is essential because the square root of negative numbers is not defined in the realm of real numbers.

Such functions are notable for their characteristic shape, often beginning at the origin, but in cases like ours, transformations such as shifts can alter their position. Understanding how this function behaves is the first step to mastering function transformations and analyzing complex mathematical expressions.

Domain and Range

When analyzing any function, determining the domain and range is a vital step. The domain consists of all possible input values (\( x \)) that will yield a valid output in the function, while the range is the set of all potential outputs (\( f(x) \)).

For the square root function \( \sqrt{x^2 + 1} \), the expression inside the square root determines the domain. Since \( x^2 + 1 \) is always positive (as squares are always positive), any real number can be an input, meaning the domain is all real numbers. Essentially, \(-\infty < x < \infty\).

The range of our specific function \( f(x) = \sqrt{x^2 + 1} - 4 \) is derived by considering the minimum value the function can output. Since the minimum value of \( \sqrt{x^2 + 1} \) is 1 (when \( x = 0 \)), we have \( \sqrt{1} - 4 = -3 \). Thus, \( f(x) \) can never output a value less than \(-3\), making the range \( [-3, \infty) \). Understanding the domain and range helps in graphing and analyzing the behavior of functions effectively.

Function Transformation

Function transformation refers to the ways in which the basic function can be modified to create new functions. This process involves altering features of the function such as shifting, stretching, compressing, or reflecting its graph.

For our example, \( f(x) = \sqrt{x^2 + 1} - 4 \), the square root part \( \sqrt{x^2 + 1} \) is the core component, while the \(-4\) signifies a vertical shift. This means every point on the function's graph gets moved down 4 units along the y-axis. Such vertical shifts do not affect the domain, but they do alter the range of the function.

In a broader context, function transformations can involve several other modifications:

• Horizontal Shifts: Move the graph left or right.
• Vertical Shifts: Move the graph up or down, as seen here.
• Reflections: Flip the graph over a particular axis.
• Stretching/Compressing: Increase or decrease the distance of points from the axis.

Understanding transformation enables one to manipulate and adapt functions to fit various situations, making it a crucial skill in algebra functions.