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Chapter 1: Problem 2

State the domain of the function defined by the given expression. $$ \sqrt{x^{2}+2} $$

Short Answer

Expert verified
The domain of \( \sqrt{x^2+2} \) is all real numbers, \((-\infty, \infty)\).

Step by step solution

01

Identify the Domain Concept

To find the domain of a function involving a square root, we need to ensure that the expression under the square root is non-negative, as the square root of a negative number is not defined in the set of real numbers.
02

Set Up the Inequality

For the function \( \sqrt{x^2+2} \), set the inequality under the square root to be greater than or equal to zero: \[ x^2 + 2 \geq 0 \].
03

Solve the Inequality

Since \( x^2 \) is always non-negative for all real numbers \( x \), and adding 2 makes \( x^2+2 \) strictly positive, this expression is always greater than zero. Thus, the inequality holds for all real \( x \).
04

Conclude the Domain

Since \( x^2 + 2 \geq 0 \) for all real \( x \), the domain of the function \( \sqrt{x^2+2} \) is all real numbers, denoted as \( (-\infty, \infty) \).

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Key Concepts

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

Square Root Function
A square root function involves an expression under the square root symbol. For the square root to be defined, the expression inside must be non-negative. This is because within the set of real numbers, the square root of a negative number isn't defined.

In mathematical terms, for a function defined as \( \sqrt{f(x)} \), where \( f(x) \) is an expression inside the root, \( f(x) \) must be greater than or equal to zero. The reason is that only non-negative results yield a real number when taking the square root.
  • Example: For the function \( \sqrt{x^2 + 2} \), we check if \( x^2 + 2 \geq 0 \).
  • Key Takeaway: Inside the square root is always evaluated for non-negativity.
Inequality Solving
Solving inequalities is crucial when determining the domain of square root functions. You set up the inequality based on the expression under the square root, ensuring it is non-negative.

Consider the function with a square root: \( \sqrt{x^2 + 2} \). To find its domain:
  • Set up inequality: \( x^2 + 2 \geq 0 \)
  • Solve inequality: Realize \( x^2 \) is always non-negative, thus \( x^2 + 2 \) is always positive.
  • Since the inequality holds true for all real numbers, the domain includes all real numbers.
Understanding inequalities allows you to determine where an expression within a function becomes undefined or defines the function completely within real numbers.
Real Numbers
The set of real numbers includes all the possible numbers that we can think of on the number line, from negative infinity to positive infinity. Real numbers encompass both rational numbers (like 1/2, 4) and irrational numbers (like \( \sqrt{2} \), \( \pi \)).

In the context of functions, particularly those involving square roots, real numbers determine the possible values that the input \( x \) can take.
  • The domain \( (-\infty, \infty) \) means that any real number is a valid input for the function.
  • For \( \sqrt{x^2 + 2} \), since the expression \( x^2 + 2 \) is always non-negative, we conclude the function's domain spans all real numbers.
This concept is fundamental when dealing with functions, ensuring that you comprehend the entire scope of real-valued solutions across different scenarios.

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Most popular questions from this chapter

Find a function \(g\) such that \(h=g \circ f\) \(h(x)=\left(x^{2}+1\right) /\left(x^{4}+2 x^{2}+3\right), f(x)=x^{2}+1\)

Suppose \(p(x)\) is a polynomial of degree \(n>1\). Let \(I(x)=x,\) and define \(f=p \circ(I+p) .\) What is deg \((f) ?\) How are the roots of \(p\) related to the roots of \(f\) ? Show that there is a polynomial \(q\) such that \(f=p \circ q .\) Illustrate with \(p(x)=x^{2}-3 x-4\)

Find a function \(g\) such that \(g \circ f=h\) \(f(x)=2 x+3, h(x)=(x+5) /(x-5)\)

Write the given polynomial as a product of irreducible polynomials of degree one or two. \(x^{4}+3 x^{2}+2\)

Graph \(y=1 /\left(1+x^{2}\right)\). Add the lower half of the circle centered at \(P=(9 / 4,3)\) with radius \(5 \sqrt{5} / 4\) to your graph. Notice that curve and the circle appear to be tangent at the point of intersection \(Q .\) What is the equation of the line through \(P\) and \(Q ?\) What is the line \(\ell\) through \(Q\) that is perpendicular to \(P Q ?\) Add the plot of \(\ell\) to your graph.
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