Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(H(x)=\sqrt{x^{2}+4}\)

Step-by-Step Solution

Answer

The domain of \(H(x)\) is all real numbers and its range is \([2, \infty)\).

Step 1: Understanding the Square Root Function

The function given is \[ H(x) = \sqrt{x^2 + 4} \]The square root function takes whatever is inside the radical (in this case, \(x^2 + 4\)) and finds its non-negative square root. Since square roots are defined for non-negative real numbers under real-number arithmetic, this has implications for the domain of the function.

Step 2: Determine the Domain

The domain of the function is determined by where the expression inside the square root is non-negative. Therefore, we need \[ x^2 + 4 \geq 0 \]Since \(x^2\) is always non-negative and adding 4 keeps the expression positive, it follows that \[ x^2 + 4 \geq 4 \]is always true for all real numbers. Therefore, the domain of \(H(x)\) is all real numbers, which can be written as \((-\infty, \infty)\).

Step 3: Evaluate the Function at a Specific Point

To better understand the function, let's evaluate it at a specific point, say \( x = 0 \).\[ H(0) = \sqrt{0^2 + 4} = \sqrt{4} = 2 \]This shows that the output of the function when \( x = 0 \) is 2. We can similarly evaluate other points to understand the behavior of the function, but for now, note that \( H(x) \geq 2 \) for all \(x\).

Step 4: Analyze the Behavior

The function \( H(x) \) involves a square root of a quadratic expression. As \( x \) increases or decreases, the function behaves as follows:- As \( x \rightarrow \infty \), the term \( x^2 \) dominates, and so \( H(x) \sim |x| \).- Similarly, as \( x \rightarrow -\infty \), \( H(x) \sim |x| \).

Step 5: Determine the Range of the Function

Since \( H(x) = \sqrt{x^2 + 4} \) and we have seen that for any \( x \), \( H(x) \geq 2 \), the minimum value occurs when \( x^2 = 0 \) (i.e., \( x = 0 \)). Therefore, the range of \( H(x) \) is \([2, \infty)\).

Key Concepts

Domain and Range of Square Root Functions

The domain of a function refers to all the possible input values (or x-values) the function can accept without causing any mathematical issues, such as dividing by zero or taking a square root of a negative number. For our function, \( H(x) = \sqrt{x^2 + 4} \), we need to make sure that the expression under the square root, \( x^2 + 4 \), is always non-negative.
This expression is non-negative for all real numbers because:

• \( x^2 \) is always non-negative since squaring any real number gives a positive result or zero.
• Adding 4 shifts this non-negative result higher, ensuring it is always positive.

Thus, the expression \( x^2 + 4 \geq 4 \) is always true, making the domain all real numbers, \(( -\infty, \infty)\). This means you can plug any real number into the function and get a valid output.
As for the range, which represents all possible output values (or y-values) of the function, we need to consider the smallest output value. Since \( x^2 \geq 0 \) and it becomes \( x^2 + 4 \geq 4\), the smallest it can be is when \( x = 0 \), giving \( H(0) = 2 \). Thus, the range is \([2, \infty)\). The function will produce values starting from 2 and going up to infinity.

Understanding Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In our problem, we are dealing with a quadratic expression inside a square root: \( x^2 + 4 \).
This particular quadratic expression is simple because it doesn't have the \( bx \) term, making it just \( x^2 + 4 \). This simplifies our calculations. Quadratics typically graph as parabolas. The graph of \( x^2 + 4 \) is a parabola that opens upwards, shifted 4 units up the y-axis due to the +4.

• \( x^2 \) is the square term, making the parabola's basic shape.
• The constant, +4, shifts this shape upwards.

This upward shift ensures that \( x^2 + 4 \) is always positive, which is crucial for square root functions because they require non-negative inputs. Understanding this helps us analyze functions involving quadratics more easily, especially when they are under a square root.

Exploring Real Numbers

Real numbers include all the numbers that can be found on the number line. This broad category involves:

• Natural numbers (1, 2, 3, ...),
• Whole numbers (0, 1, 2, 3, ...),
• Integers (..., -2, -1, 0, 1, 2, ...),
• Rational numbers (fractions or numbers that can be expressed as a quotient of integers),
• Irrational numbers (numbers that cannot be written as simple fractions, like \( \sqrt{2} \) or \( \pi \)).

In our exercise, when we talk about the domain being all real numbers, \(( -\infty, \infty)\), it means you can choose any of these numbers to substitute in your function \( H(x) = \sqrt{x^2 + 4} \). Real numbers encompass everything from simple integers to complex irrational numbers, and understanding this set helps to see why \( x^2 + 4 \) is always non-negative.
The real numbers ensure that the function is well-defined across an entire continuum of possibilities, giving \( H(x) \) its unrestricted domain. It’s what makes mathematical solutions robust and widely applicable, reflecting the broad scope of real-world situations.