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Chapter 7: Problem 109

Let \(f(x)=3 x-1\) and \(g(x)=\frac{1}{x}\). Find the domain of \(g\).

Short Answer

Expert verified
The domain of \( g(x) \) is all real numbers except 0.

Step by step solution

01

- Define the Domain of a Function

The domain of a function refers to all the possible input values (x-values) that the function can accept without leading to undefined or invalid mathematical operations.
02

- Identify the Function

The given function is \( g(x) = \frac{1}{x} \). This is a rational function.
03

- Determine when the Function is Undefined

A rational function is undefined when its denominator is zero. For \( g(x) = \frac{1}{x} \), the function will be undefined when its denominator (x) is zero.
04

- Identify the Excluded Value

Set the denominator to zero and solve: \(x = 0\). Therefore, \( g(x) \) is undefined at \( x = 0 \).
05

- State the Domain

The domain of \( g(x) \) includes all real numbers except 0. In interval notation, this is written as \( (-\forall, 0) \bigcup (0, \forall) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational function
A rational function is a type of function represented as the ratio of two polynomials. For example, the given function \( g(x) = \frac{1}{x} \) is a simple rational function because it can be expressed as a polynomial divided by another polynomial.

In general, a rational function can be written as \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. The key feature of rational functions is that they can exhibit undefined values where the denominator, \( Q(x) \), is zero.
undefined values
Undefined values occur in rational functions when the denominator equals zero. Mathematically, division by zero is undefined, leading to gaps or asymptotes in the graph of the function.

For the function \( g(x) = \frac{1}{x} \), we find the undefined value by setting the denominator equal to zero:
\[ x = 0 \ 1 = 0 \]
Since division by zero is not possible, \( g(x) \) does not exist at \( x = 0 \). Hence, \( g(x) \) is undefined at this point.

It's important to identify these values as they help determine the function's domain.
interval notation
Interval notation is a concise way of expressing the set of all possible values (domain) for which the function is defined.

For the function \( g(x) = \frac{1}{x} \), the domain includes all real numbers except 0. We use interval notation to express this domain.

Using interval notation, we write:
\( (-\infty, 0) \cup (0, \infty) \)
This notation signifies that \( g(x) \) is defined for all real numbers except at \( x = 0 \). This way of representation is efficient and allows for clear communication of the function's domain.

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