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Chapter 1: Problem 60

Find the domain of the function. $$f(x)=\frac{x+2}{\sqrt{x-10}}$$

Short Answer

Expert verified
The domain of the function \(f(x) = \frac{x+2}{\sqrt{x-10}}\) is \(x > 10\).

Step by step solution

01

Identify where the function is undefined

To understand the domain of the function, one needs to know where it is undefined. The function is undefined where the denominator is zero or when there is an even root of a negative number. Identify these points.
02

Denominator Cannot be Zero

In the given function \(f(x) = \frac{x+2}{\sqrt{x-10}}\), the denominator portion is \(\sqrt{x-10}\). As a rule, the denominator cannot be zero because division by zero is undefined. Solve the equation \(x-10 = 0\) to find out this point.
03

Solve for x where \(x-10 = 0\)

Solve for x when \(x-10 = 0\). The solution is \(x = 10\). Hence the function is undefined at \(x = 10\).
04

Identify for which x the \(x-10\) is negative

In the denominator, there is a square root. An even root of a negative number is also undefined. So we need to figure out for which x, \(x-10\) is negative. Solve \(x-10 < 0\) to find this range.
05

Solve for x where \(x-10 < 0\)

Solve for x when \(x-10 < 0\). The solution is \(x < 10\). Therefore, the function is also undefined for all values where \(x < 10\). It means the function is undefined from negative infinity to 10.
06

Identify the Domain

Identify the domain as the range of x where the function is defined. The function is undefined where \(x < 10\), or where \(x = 10\), which means it will be defined for \(x > 10\).

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