Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=\frac{3 x-4}{-x}$$

Step-by-Step Solution

Answer

The graph of the given function \(g(x) = \frac{3x - 4}{-x}\) will reveal its domain is all real numbers except x = 0, it has a vertical asymptote at x = 0, and a horizontal asymptote at y = -3.

Step 1: Simplifying the Function

The given function, \(g(x) = \frac{3x - 4}{-x}\), can be simplified to \(g(x) = -3 + \frac{4}{x}\) by dividing each term of the numerator by the denominator, -x.

Step 2: Determine the Domain

The function will be undefined when the denominator is equal to zero. Therefore, by solving -x = 0, we get x = 0. Hence, the domain of the function \(g(x)\) is all real numbers except for x = 0.

Step 3: Identify Vertical Asymptotes

Vertical asymptotes occur where the function is undefined. From the previous step, we know that x = 0 makes the denominator 0 and thus the function is undefined. Hence, x = 0 is a vertical asymptote.

Step 4: Identify Horizontal Asymptotes

A horizontal asymptote is a value that the function approaches as x goes to either positive or negative infinity. Here, as x gets very large or very small, the term \(\frac{4}{x}\) goes closer to 0, and the function gets closer and closer to -3. Thus, the horizontal asymptote is y = -3.

Step 5: Graphing the Function

Plot the function \(g(x) = -3 + \frac{4}{x}\) using a graphing utility taking into consideration the domain, the vertical asymptote at x = 0 and the horizontal asymptote at y= -3

Key Concepts

Domain of a Function

The domain of a function determines all possible input values (or "x" values) for which the function is defined, meaning it doesn't involve any mathematical errors like division by zero. For the given function, \(g(x) = \frac{3x - 4}{-x}\), the domain can be found by identifying values of \(x\) that make the denominator zero as this will make the function undefined. Solving -\(x = 0\) shows that \(x = 0\) would make the denominator zero. Thus, the domain of \(g(x)\) is all real numbers except \(x = 0\).

In general, when finding the domain of rational functions (functions that are ratios of polynomials), ensure that the denominator does not equal zero. This rule helps avoid undefined operations within the function.

Vertical Asymptotes

Vertical asymptotes are vertical lines on a graph that the function approaches but never touches or crosses, often symbolizing values where the function becomes undefined. For the function \(g(x) = \frac{3x - 4}{-x}\), a vertical asymptote occurs at \(x = 0\), as this makes the denominator zero, causing the function to be undefined.

If you look at the graph in the vicinity of \(x = 0\), you'll notice that as \(x\) gets closer to zero from either side, the values of \(g(x)\) increase or decrease without bound. This steep, undefined behavior is characteristic of a vertical asymptote.

To find vertical asymptotes in rational functions, solve for values of \(x\) that make the denominator zero, keeping in mind any variables from the simplified form of the function.

Horizontal Asymptotes

Horizontal asymptotes describe what happens to \(f(x)\) as \(x\) moves towards negative or positive infinity. They represent values that the function approaches but never actually reach as \(x\) becomes very large or small. In \(g(x) = -3 + \frac{4}{x}\), as \(x\) approaches infinity, the term \(\frac{4}{x}\) diminishes towards \(0\). Hence, \(g(x)\) approaches \(-3\), making \(y = -3\) the horizontal asymptote.

Horizontal asymptotes can be determined for rational functions by considering the degrees of the polynomial in the numerator compared to the polynomial in the denominator. However, in some rational functions like this one, simplification reveals the behavior directly by showing the constant term the function approaches as \(x\) gets very large.