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Chapter 2: Problem 115

What is the domain of the function \((3+x)^{1 / 4} ?\)

Short Answer

Expert verified
The domain of the function \(f(x) = (3+x)^{\frac{1}{4}}\) is all real numbers x such that \(x \geq -3\).

Step by step solution

01

Identify the Function and its Inner Expression

The given function is \(f(x) = (3+x)^{\frac{1}{4}}\). The expression inside the fourth root is \(3+x\).
02

Determine the Constraints on the Inner Expression

Since we are dealing with a fourth root, we know that the expression inside the root must be non-negative. That is, \(3+x \geq 0\).
03

Solve for the Domain

Now, we will solve the inequality \(3+x \geq 0\) to determine the valid values of x. 1. Subtract 3 from both sides of the inequality: \(x \geq -3\) So, the domain of the function \(f(x) = (3+x)^{\frac{1}{4}}\) includes all real numbers x such that \(x \geq -3\).

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Most popular questions from this chapter

Suppose \(q\) is a polynomial of degree 4 such that $$ \begin{array}{r} q(0)=-1 . \text { Define } p \text { by } \\ \qquad p(x)=x^{5}+q(x) . \end{array} $$ Explain why \(p\) has a zero on the interval \((0, \infty)\).

Show that if \(p\) and \(q\) are nonzero polynomials with \(\operatorname{deg} p<\operatorname{deg} q,\) then \(\operatorname{deg}(p+q)=\operatorname{deg} q\).

Find all real numbers \(x\) such that $$ x^{6}-3 x^{3}-10=0 $$.

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Give an example of a polynomial \(p\) of degree 8 such that \(p(2)=3\) and \(p(x) \geq 3\) for all real numbers \(x\).
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