Question

Find the domain of each function. \(G(x)=\frac{x+4}{x^{3}-4 x}\)

Short answer

The domain is all real numbers except \( x = 0 \), \( x = 2 \), and \( x = -2 \).

Step-by-step solution

Identify the Function

Given the function is: \[ G(x) = \frac{x + 4}{x^3 - 4x} \]We need to find the domain of this function.

Find the Denominator

To determine the domain, focus on the denominator to make sure it doesn’t equal zero. For the given function, the denominator is \[ x^3 - 4x \].

Identify Values that Make the Denominator Zero

Set the denominator equal to zero:\[ x^3 - 4x = 0 \]Factor the equation to find values of x that will make this true:\[ x(x^2 - 4) = 0 \]\[ x(x - 2)(x + 2) = 0 \].

Solve for x

Set each factor equal to zero and solve:\[ x = 0 \]\[ x - 2 = 0 \] which gives \[ x = 2 \]\[ x + 2 = 0 \] which gives \[ x = -2 \].Thus, the denominator is zero when \( x = 0 \), \( x = 2 \), or \( x = -2 \).

Define the Domain

The domain of the function \( G(x) \) includes all real numbers except those that make the denominator zero. Therefore, the domain is all real numbers except \( x = 0 \), \( x = 2 \), and \( x = -2 \).

Key concepts

Rational Functions

Rational functions might sound complicated, but they are very common in math.
These functions represent the ratio of two polynomials.
For example, in the function given in the exercise, we see that the numerator is a polynomial: \(x + 4\)
and the denominator is another polynomial: \[x^3 - 4x\].
The main idea is to understand that a rational function is simply a fraction where both the numerator and the denominator are polynomials.
Understanding this will help clarify why and how we need to find the denominator's values that make it zero.
When the denominator is zero, the function is undefined.
This is because division by zero is not possible in mathematics.
Therefore, identifying these values is crucial in determining the function's domain.

Factoring Polynomials

Factoring polynomials is a key step in finding the domain of rational functions.
In our example, the polynomial in the denominator \(x^3 - 4x\)
can be factored to more easily find the values of x that make the denominator zero.
First, we factor out the common term \(x\):
\begin{aligned} x^3 - 4x & = x(x^2 - 4) \ & = x(x - 2)(x + 2) \end{aligned}
Looking at this step-by-step:

• We start by pulling out the common factor \(x\). This makes it easier to work with the remaining terms.
• We notice that \(x^2 - 4\) is a difference of squares and can be factored into \( (x - 2)(x + 2) \).
• Once factored completely, it becomes clear that the polynomial can be written as: \[ x(x - 2)(x + 2) \]

Factoring makes it much easier to find the excluded values because now we can set each factor equal to zero and solve for x.

Excluded Values

The excluded values of a function are the values that make the denominator zero.
For the function given, we identified that the factored form of the denominator is \( x(x - 2)(x + 2) \).
To find the excluded values where the function is undefined, we set each factor to zero:

• \ x = 0 \
• \ x - 2 = 0 \ which gives \ x = 2 \
• \ x + 2 = 0 \ which gives \ x = -2 \

Therefore, the excluded values are \ x = 0 \, \ x = 2 \, and \ x = -2 \. These are the points where the denominator becomes zero, making the function undefined.
To summarise, the domain of the given function \[ G(x) = \frac{x + 4}{x^3 - 4x} \] is all real numbers except \ x = 0 \, \ x = 2 \, and \ x = -2 \.