Problem 82 - Calculus: Early Transcendentals

Question

Use the continuity of the absolute value function (Exercise 78 ) to determine the interval(s) on which the following functions are continuous. $$h(x)=\left|x^{2}+2 x+5\right|+\sqrt{x}$$

Step-by-Step Solution

Answer

Answer: The function $$h(x)$$ is continuous on the interval \([0, \infty)\).

Step 1: Find the domain for the absolute value function

The absolute value function $$|x^2 + 2x + 5|$$ is continuous for all real numbers because it is a polynomial function inside an absolute value function. Hence, its domain is \((-\infty, \infty)\).

Step 2: Find the domain for the square root function

The square root function $$\sqrt{x}$$ is continuous for all non-negative values of x since it is not defined for negative values. Its domain is \([0, \infty)\).

Step 3: Find the intersection of the domains

To find the interval where the function $$h(x)$$ is continuous, we can find the intersection of the domains of the component functions. The intersection of \((-\infty, \infty)\) and \([0, \infty)\) is \([0, \infty)\).

Step 4: Answer the question

Hence, the function $$h(x) = \left|x^2 + 2x + 5\right| + \sqrt{x}$$ is continuous on the interval \([0, \infty)\).

Key Concepts

Absolute Value Function

The absolute value function, typically represented as \( |x| \), takes any real number \( x \) and outputs its non-negative value. It essentially reflects the negative part of its graph across the x-axis, creating a \( V \)-shape. This means that whether the input is negative or positive, the output is always non-negative. Mathematically, \( |x| = x \) if \( x \ge 0 \) and \( |x| = -x \) if \( x < 0 \).

Due to its definition, the absolute value function modifies other functions inside the absolute value signs. Polynomial functions, such as \( x^2 + 2x + 5 \), are continuous for all real numbers. When placed inside an absolute value function, this continuity is preserved across the entire set of real numbers, making the domain \( (-\textrm{\infty}, \textrm{\infty}) \).

Example in Practice

For \( h(x) = |x^2 + 2x + 5| \), the function will graph as the same parabola \( x^2 + 2x + 5 \) where it is positive, and as a reflected parabola where it would be negative. This ensures that \( h(x) \) is never negative for any real number input.

Domain of a Function

The domain of a function comprises all the possible input values (typically \( x \)-values) for which the function is defined and yields a real number output. For different types of functions, the domain can vary widely. For instance, polynomials have all real numbers in their domain, while square root functions are only defined for non-negative inputs.

To determine a function's domain, we must consider any restrictions from operations within the function, such as division (cannot divide by zero) and even roots (cannot take the square root of a negative number in the real number system).

Identifying Domain Restrictions

When examining \( h(x) = \sqrt{x} \), the square root imposes the restriction that \( x \) must be greater than or equal to zero, giving us the domain \( [0, \textrm{\infty}) \). It's critical to identify such restrictions to ascertain the intervals over which the function is defined and consequently continuous.

Square Root Function

The square root function, symbolized as \( \sqrt{x} \), assigns to every non-negative number \( x \) its principal square root. The graph of this function is part of a parabola lying on its side, with the 'pointy end' at the origin (0,0), extending infinitely towards the right. By definition, square roots cannot be taken of negative numbers in the real number system, as they yield complex numbers.

For the function \( h(x) = \sqrt{x} \), the domain is confined to \( x \ge 0 \), because for negative values of \( x \), the square root is not real. Thus, the square root function is continuous on its entire domain of \( [0, \textrm{\infty}) \).

Continuous Yet Restrictive

While square root functions are continuous, they inherently restrict the overall domain of composite functions like \( h(x) = |x^2 + 2x + 5| + \sqrt{x} \), as we need to consider only the inputs where the square root is defined.

Intersection of Domains

When dealing with functions comprised of multiple parts, such as \( h(x) = |x^2 + 2x + 5| + \sqrt{x} \), we need to find a common range of inputs where both parts are valid. This is known as finding the intersection of their domains. The intersection is simply the set of all values that belong to both domains simultaneously.

In our case, to find where \( h(x) \) is continuous, we intersect the domain \( (-\textrm{\infty}, \textrm{\infty}) \) of the absolute value function with the domain \( [0, \textrm{\infty}) \) of the square root function. The result is the latter domain — \( [0, \textrm{\infty}) \) — since the square root imposes the stricter requirement that \( x \) must be non-negative.

Finding Common Ground

Intersecting domains help us pinpoint the exact interval on which a composite function can operate without breaking 'mathematical rules', ensuring the function behaves as expected for all inputs in that range.

Continuous Interval

A continuous interval for a function implies that there are no breaks, jumps, or holes in the graph of the function for any inputs within that interval. Continuity is a key concept in calculus and real-world modeling because it ensures smoothness in the function's behavior over a range of inputs.

In the context of our function \( h(x) = |x^2 + 2x + 5| + \sqrt{x} \), after identifying the intersection of domains, we find that it is continuous over the interval \( [0, \textrm{\infty}) \). This means that for every point \( x \) within that range, the function smoothly transitions from one \( x \)-value to the next with no unexpected variations.

Why Continuity Matters

Understanding the continuous intervals of functions allows students to better anticipate the behavior of functions, particularly in problem-solving scenarios where real-life phenomena are being modeled, such as physics, biology, or economics.