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Chapter 2: Problem 30

Identify the set of values \(x\) for which \(y\) will be a real number. $$ y=\sqrt{2.2-x} $$

Short Answer

Expert verified
The set of values for \(x\) is \(x \leq 2.2\).

Step by step solution

01

Set the radicand non-negative

Identify that the expression inside the square root, known as the radicand, must be non-negative for the result to be a real number. The radicand here is given by \(2.2 - x\). Therefore, we set up the inequality \(2.2 - x \geq 0\).
02

Solve the inequality

Solve the inequality from Step 1 to find the values of \(x\) for which \(2.2 - x \geq 0\). Subtract \(2.2\) from both sides to get \(-x \geq -2.2\). Then, multiply both sides of the inequality by \(-1\) (remembering to reverse the inequality sign) to obtain \(x \leq 2.2\).

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Key Concepts

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

inequality
An inequality is a mathematical statement that expresses the relationship of difference between two values. It uses symbols like: <, >, 鈮�, 鈮�.

For example, in your problem, we use the inequality symbol (鈮�) to ensure that the expression inside the square root (the radicand) is non-negative. This is important because the square root of a negative number does not yield a real number.

By setting up the inequality \(2.2 - x \geq 0\), we ensure that x-values are restricted to numbers that keep the radicand of the square root non-negative.
real number
A real number is any value along the continuous number line. This includes all integers, fractions, and decimals. Real numbers can be either positive, negative, or zero.

To determine when the given function \( y = \sqrt{2.2 - x} \) yields a real number, the radicand (2.2 - x) must not be negative. This ensures the result of the square root function is a real number.

If the radicand were negative, the result would be an imaginary number, which would mean y is not a real number. The goal is to find the values of x such that y remains a real number.
square root
The square root is a mathematical function that, when applied to a non-negative number, gives a value that, when multiplied by itself, results in the original number. It is represented by the symbol 鈭�.

In our exercise, \( y = \sqrt{2.2 - x} \), we must ensure that \(2.2 - x\) is non-negative because the square root of a negative number is not a real number.

So, by setting the radicand \(2.2 - x\) greater than or equal to 0, we can find the valid domain of x that keeps the square root function within the realm of real numbers.
radicand
The radicand is the number or expression inside the square root symbol. In the problem \( y = \sqrt{2.2 - x}\), the radicand is \( 2.2 - x \).

For the square root function to return real numbers, the radicand must be non-negative. This means it must be zero or a positive number. By solving the inequality \( 2.2 - x \geq 0 \), we ensure that the radicand remains non-negative, so the function produces valid real number outputs for y.

Thus, we determined that \[ x \leq 2.2 \] to keep the radicand non-negative. This means that any value of x less than or equal to 2.2 will keep the radicand (2.2 - x) within the range required for y to be a real number.

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Most popular questions from this chapter

Evaluate the function for the given values of \(x\). \(g(x)=\left\\{\begin{array}{ll}-2|x|-3 & \text { for } x \leq-2 \\ 5 x+6 & \text { for }-2

a. Graph \(f(x)=|x|\) for \(x<0\). b. Graph \(g(x)=\sqrt{x}\) for \(x \geq 0\). c. Graph \(h(x)=\left\\{\begin{array}{ll}|x| & \text { for } x<0 \\ \sqrt{x} & \text { for } x \geq 0\end{array}\right.\)

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a. Graph \(a(x)=x\) for \(x<1\). b. Graph \(b(x)=\sqrt{x-1}\) for \(x \geq 1\). c. Graph \(c(x)=\left\\{\begin{array}{ll}x & \text { for } x<1 \\ \sqrt{x-1} & \text { for } x \geq 1\end{array}\right.\)

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