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Chapter 2: Problem 3

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

Short Answer

Expert verified
The domain of the function \(g(x)=\frac{3}{x-4}\) is all real numbers except x=4.

Step by step solution

01

Identify the function type

The function given, \(g(x)=\frac{3}{x-4}\), is a rational function because it is a ratio of two polynomials.
02

Understanding domain restrictions on a rational function

Rational functions will be undefined when the denominator equals zero. This is because division by zero is undefined in mathematics.
03

Identify when the function is undefined

To find when the function is undefined, you set the denominator equal to zero and solve for \(x\). In this case, setting \(x-4=0\), we get the solution \(x=4\). This means the function is undefined when \(x=4\).
04

Find the domain

So, to find the domain of the function, we need all real numbers except where the function is undefined. Hence, the domain of the function is 'All real x, \(x \ne 4\)'.

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Most popular questions from this chapter

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-3)^{2}+(y+1)^{2} &=9 \\ y &=x-1 \end{aligned} $$

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-4,0), r=10 $$

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ f(x)=x^{3}-6 x^{2}+9 x+1 $$

What is a piecewise function?

Does \((x-3)^{2}+(y-5)^{2}=-25\) represent the equation of a circle? What sort of set is the graph of this equation?
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