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Chapter 1: Problem 66

Find the domain of the function. $$f(x)=\frac{\sqrt{x+6}}{6+x}$$

Short Answer

Expert verified
The domain of the function \( f(x) = \frac{\sqrt{x+6}}{6+x} \) is \(x > -6\).

Step by step solution

01

Find the Condition for the Square Root

The square root must have a non-negative argument. So \(x+6 >= 0\) which implies that \(x >= -6\). So, all real numbers greater than or equal to -6 will not make the square root undefined.
02

Find the Condition for the Denominator

The denominator of the fraction cannot equal zero because division by zero is undefined. So, \(6+x != 0\) which simplifies to \(x != -6\). Thus, -6 should be excluded from the domain.
03

Combine the Conditions

The conditions derived from the square root and the denominator need to be combined. This means that we are looking for all real numbers which are greater than or equal to -6, but not equal to -6. The domain of the function, therefore, is \(x > -6\).

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Most popular questions from this chapter

Your wage is \(\$ 12.00\) per hour plus \(\$ 0.55\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced \(x\) is \(y=12+0.55 x.\) (a) Find the inverse function. What does each variable in the inverse function represent? (b) Use a graphing utility to graph the function and its inverse function. (c) Use the trace feature of the graphing utility to find the hourly wage when 9 units are produced per hour. (d) Use the trace feature of the graphing utility to find the number of units produced per hour when your hourly wage is \(\$ 21.35\)

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Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$

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