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Chapter 2: Problem 116

What is the domain of the function \(\left(1+x^{2}\right)^{1 / 8} ?\)

Short Answer

Expert verified
The domain of the function \(\left(1+x^{2}\right)^{1 / 8}\) is all real numbers, expressed as the interval notation \((-\infty, \infty)\) or in set notation \(\{x: x \in \mathbb{R}\}\), since \(1+x^2\) is always non-negative for all x.

Step by step solution

01

Identify the expression inside the root

The expression inside the 8th root is \(1+x^2\). Our goal is to find values of x such that this expression is non-negative.
02

Determine when the expression inside the root is non-negative

The expression is \(1+x^2\). We want to find when this is greater than or equal to 0. Since \(x^2\) is always non-negative (meaning \(x^2 \ge 0\) for all x), and adding 1 to a non-negative number will still yield a non-negative number, then \(1+x^2\) will also always be non-negative.
03

Write the domain of the function

Since \(1+x^2\) is always non-negative for all x, the domain of the function \(\left(1+x^{2}\right)^{1 / 8}\) is all real numbers, which can be written as the interval notation \((-\infty, \infty)\) or in set notation \(\{x: x \in \mathbb{R}\}\).

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Most popular questions from this chapter

Explain why the composition of two rational functions is a rational function.

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ \frac{s(1+x)-s(1)}{x} $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (3 r-2 s)(x) $$

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Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$
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