Problem 107 - Brief Applied Calculus

Question

For each statement, state in words the values of \(x\) for which each exponential expression is defined. $$ x^{1 / 2} $$

Step-by-Step Solution

Answer

The expression is defined for all \(x \geq 0\).

Step 1: Understanding Exponential Terms

The expression is written as \(x^{1/2}\). This is synonymous with the square root of \(x\), often written as \(\sqrt{x}\). We need to determine the conditions under which a square root is defined.

Step 2: Square Root Definition

A square root is only defined for non-negative numbers. Therefore, for the square root, which is \(x^{1/2}\), to be defined, \(x\) must be greater than or equal to zero.

Step 3: Conclusion

Since \(x^{1/2}\) represents the square root of \(x\), the expression is defined for all values where \(x \geq 0\).

Key Concepts

Square Roots

Square roots are mathematical operations used to find a number which, when multiplied by itself, yields the original number. In algebra, the square root of a number is represented by a radical sign, like this: \( \sqrt{x} \). Alternatively, it can be expressed with an exponent: \( x^{1/2} \).

It's important to remember:

• Square roots are defined only for non-negative numbers. This means that the expression \( \sqrt{x} \) is valid and yields a real number only if \( x \) is greater than or equal to zero.
• For example, \( \sqrt{4} = 2 \), because \( 2 \times 2 = 4 \), and in general, \( \sqrt{x} = x^{1/2} \) represents the same value.

Switching between these notations allows mathematicians to write complex expressions cleanly and perform calculations more flexibly.

Domain of a Function

The domain of a function refers to the set of all possible input values (usually \( x \) values) that are allowed in the function. When dealing with functions that involve square roots, defining the domain is crucial, as it tells us which values of \( x \) we can use without encountering mathematical errors.

For a function involving a square root, such as \( f(x) = \sqrt{x} \):

• We must ensure that \( x \) is non-negative, i.e., \( x \geq 0 \), since square roots of negative numbers are not defined in the set of real numbers.
• The domain of \( f(x) = \sqrt{x} \) can thus be said to include all non-negative numbers.

So, when faced with any function involving square roots, always check the input values to ensure they fall within the valid range; this way, you're determining its domain correctly.

Non-negative Numbers

Non-negative numbers are numbers that are either greater than or equal to zero. They include all positive numbers, as well as zero itself. Understanding non-negative numbers is essential in ensuring the validity of operations like square roots.

Here's what you need to know:

• Non-negative numbers are written as \( x \geq 0 \). This is crucial in solving problems involving the square roots since square roots are only defined for these inputs.
• Consider that when \( x \geq 0 \), both \( \sqrt{x} \) and \( x^{1/2} \) will yield valid results.
• Thus, when dealing with equations like \( x^{1/2} \), you must confirm that \( x \) is a non-negative number to make mathematical sense.

Mastering the concept of non-negative numbers ensures clarity and accuracy in dealing with a range of mathematical problems.