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Chapter 2: Problem 16

Find the domain of each function. $$f(x)=\frac{1}{\frac{4}{x-2}-3}$$

Short Answer

Expert verified
The domain of this function is all real numbers except \( x=\frac{10}{3} \)

Step by step solution

01

Identify the Denominator

First, identify the denominator of the function. It is necessary to set the denominator not equal to zero. Here, the denominator is \( \frac{4}{x-2}-3 \)
02

Set the Denominator not Equal to Zero

To find values that are not in the domain of the function, set the denominator not equal to zero and solve for \( x \):\(\frac{4}{x-2}-3 \neq 0\)
03

Isolate the Fraction

Add 3 to both sides of the equation:\(\frac{4}{x-2} \neq 3\)
04

Solve for x

Finally, find the x-value that would cause the denominator to be zero. Multiply both sides of the equation by \( x - 2 \):\(4 \neq 3x - 6\)Then, add 6 to both sides and divide all terms by 3:\(x \neq \frac{10}{3}\)

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Most popular questions from this chapter

will help you prepare for the material covered in the next section. $$ \text { If }\left(x_{1}, y_{1}\right)=(-3,1) \text { and }\left(x_{2}, y_{2}\right)=(-2,4), \text { find } \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$

What is the graph of a function?

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+3)^{2}+(y-2)^{2}=4 $$

Explain how the vertical line test is used to determine whether a graph represents a function.

What is a relation? Describe what is meant by its domain and its range.
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