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Chapter 1: Problem 107

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

Conclusion

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=4 x^{2}-5 x+3 $$

\(71-72 .\) GENERAL: Stopping Distance A car traveling at speed \(v\) miles per hour on a dry road should be able to come to a full stop in a distance of $$ D(v)=0.055 v^{2}+1.1 v \text { feet } $$ Find the stopping distance required for a car traveling at: \(40 \mathrm{mph}\).

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{3}{x} $$

BUSINESS: Cost Functions A lumberyard will deliver wood for \(\$ 4\) per board foot plus a delivery charge of \(\$ 20\). Find a function \(C(x)\) for the cost of having \(x\) board feet of lumber delivered.

BIOMEDICAL: Cell Growth The number of cells in a culture after \(t\) days is given by \(N(t)=200+50 t^{2}\). Find the size of the culture after: a. 2 days. b. 10 days.
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