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The domain of a function defines all the possible input values (or x-values) for which the function is valid. For rational functions, which are expressed in the form \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials, the domain excludes any x-values that make the denominator zero. This is because division by zero is undefined in mathematics.

To find the domain of the function \(g(x) = \frac{4x - 20}{3x - 18}\), we focus on the denominator \(3x - 18\). Set the denominator equal to zero and solve for \(x\):

• \(3x - 18 = 0\)
• Solve for \(x\): \(3x = 18\)
• \(x = 6\)

Thus, \(x = 6\) is not part of the domain, as it would cause division by zero.
The domain of \(g(x)\) is written in interval notation as \((-fty, 6) \cup (6, \infty)\), meaning all real numbers except \(x = 6\).

The range of a function is the set of all possible output values (or y-values). For rational functions, the range is often influenced by any horizontal asymptotes that the function may have.

In the given function \(g(x) = \frac{4x - 20}{3x - 18}\), the range can be determined by considering its behavior as \(x\) becomes very large or very small. Plus, it’s essential to identify any horizontal asymptotes because they indicate values that \(y\) will approach but never actually reach.

With a rational function of the form \(\frac{ax + b}{cx + d}\), the horizontal asymptote can be found by the leading coefficients of the numerator and the denominator.

• The horizontal asymptote is \( y = \frac{a}{c} \) or \( y = \frac{4}{3} \)

This asymptote tells us that as \(x\) approaches infinity or negative infinity, the value of the function approaches \(\frac{4}{3}\).
Therefore, the range of \(g(x)\) is \((-fty, \frac{4}{3}) \cup (\frac{4}{3}, fty)\), which means all real numbers except \(y = \frac{4}{3}\).

A horizontal asymptote in a function is a horizontal line that the graph of the function approaches as \(x\) goes to positive or negative infinity. For rational functions like \(g(x) = \frac{4x - 20}{3x - 18}\), horizontal asymptotes are determined by comparing the degrees of the polynomials in the numerator and the denominator.

If both polynomials are of the same degree, the horizontal asymptote can be found by dividing the leading coefficients:

• For \(g(x)\), both the numerator \(4x - 20\) and the denominator \(3x - 18\) have degree 1 (due to the \(x\) term).
• The horizontal asymptote is given by \( y = \frac{4}{3} \).

This means that no matter how large or small \(x\) becomes, the function will get closer and closer to this y-value, but it won't actually equal \(\frac{4}{3}\).
Understanding horizontal asymptotes is crucial in graphing rational functions because they help describe the behavior of the function at extreme values of \(x\).
For \(g(x)\), the horizontal asymptote \( y = \frac{4}{3} \) guides the range, suggesting that the function's output values will never be \(\frac{4}{3}\).