Problem 39 - Algebra 1

Question

Find the domain of the function. $$y=0.2 \sqrt{x}$$

Step-by-Step Solution

Answer

The domain of the function \(y=0.2 \sqrt{x}\) is \([0, \infty)\).

Step 1: Understanding the Square Root Function

The primary rule to consider while defining the domain of a square root function is that the term under the square root cannot be negative. This is due to the property of real numbers that square root of a negative number is undefined.

Step 2: Define the Domain

The function is \(y=0.2 \sqrt{x}\). The term under the square root is 'x'. Since it can't be negative, this implies that \(x\) must be greater than or equal to 0. Thus, the domain of the function are all real numbers that are greater than or equal to zero, often written as \([0, \infty)\).

Key Concepts

Square Root Function

The square root function is a special type of mathematical function that gets its name from involving the square root operation. When we see something like \(\sqrt{x}\), it means we're looking for a number which, when multiplied by itself, results in \(x\). This operation has unique characteristics:

• It transforms a number into its square root, simplifying expressions, or reducing dimensions in applied contexts like geometry.
• It can only be applied to non-negative numbers in the realm of real numbers, meaning \(x\) must be zero or greater, as negative numbers don't have real square roots.
• The square root function has a graph that starts at the origin \((0,0)\) and sweeps upwards to the right, showing a slow growth as \(x\) increases.

Recognizing a square root function you’ll often see it written with constants multiplying the \(\sqrt{x}\), like \(0.2\sqrt{x}\), which adjusts its steepness but not its basic shape or domain.

Real Numbers

Real numbers are a cornerstone of mathematics and form a complete, ordered field that includes both rational and irrational numbers. Unlike integers or natural numbers, real numbers can represent a more comprehensive range including all possible measurements.

• Real numbers can be positive, negative, or zero, covering fractions, whole numbers, and decimals.
• They are used to measure continuous quantities like length and height and can fill gaps between integers.
• In terms of functions, the "real" aspect ensures we are working with numbers that are physically and mathematically sensible without imaginary components.

Understanding real numbers is particularly important when dealing with domain constraints in functions like square roots, where only non-negative components make sense.

Domain Restriction

Domain restriction is a fundamental concept in algebra and calculus where we specify the "input" values (or x-values) a function can accept. Not every mathematical rule applies universally without limits, and domain restriction ensures we stay within acceptable bounds for real-world applications.

• The domain of a function is the set of all possible input values that will produce a valid output.
• When we have a square root function, like \(0.2\sqrt{x}\), the domain restriction means \(x\) cannot be negative because we cannot take the square root of a negative number without leaving the realm of real numbers.
• For such functions, the domain is restricted to \([0, \infty)\), meaning all non-negative real numbers.

Knowing how to define and apply domain restrictions helps ensure mathematical operations remain valid and interpretable, especially in applications and modeling.