Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{2 x-7}{3 x+2}$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers except \( -2/3 \). The function has a vertical asymptote at \( x = -2/3 \) and a horizontal asymptote at \( y = 2/3 \).

Step 1: Find the domain of the function

To find the domain, we set the denominator equal to zero and solve for 'x'. \n \( 3x + 2 = 0 \)\n --> Subtract 2 from both sides: \( 3x = -2 \)\n --> Divide both sides by 3: \( x = -2/3 \)\n So the denominator is zero at \( x = -2/3 \). This is the value that we should exclude from the domain, as you cannot divide by zero in math. Hence, the domain of the function \( f(x) \) is all real numbers except \( -2/3 \).

Step 2: Identify the vertical asymptote

In a rational function such as this one, vertical asymptotes occur when the denominator can be zero but the numerator cannot. From the domain calculation, we already know that the denominator is zero at \( x = -2/3 \). At these values, the function is undefined. Therefore, the function has a vertical asymptote at \( x = -2/3 \).

Step 3: Identify the horizontal asymptote

To find the horizontal asymptote of a rational function, it's necessary to compare the degrees of the polynomial in the numerator and the polynomial in the denominator. In this function, both the numerator and the denominator are first-degree polynomials, so the degrees are equal. In such a situation, the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator. The leading coefficient of the numerator is 2 and for the denominator it is 3, so the ratio is \( 2/3 \). Hence, the line \( y = 2/3 \) is a horizontal asymptote to this function.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells us where the function is defined or valid. In rational functions like \( f(x) = \frac{2x - 7}{3x + 2} \), we focus on the denominator. Since division by zero is undefined in mathematics, we must find any values of \( x \) that make the denominator zero.

To find these values, set the denominator equal to zero and solve for \( x \):

• Start with \( 3x + 2 = 0 \)
• Subtract 2 from both sides: \( 3x = -2 \)
• Divide by 3: \( x = -\frac{2}{3} \)

Thus, \( x = -\frac{2}{3} \) is the value that makes the denominator zero, so it must be excluded from the domain. Therefore, the domain of \( f(x) \) is all real numbers except \( -\frac{2}{3} \). This ensures the function remains valid and well-defined.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values that make the denominator zero while the numerator is non-zero. These are points where the function approaches infinity, either positively or negatively, and the graph will shoot up or down.

For \( f(x) = \frac{2x - 7}{3x + 2} \), we already know from the domain calculation that when \( x = -\frac{2}{3} \), the denominator is zero. Since the numerator \( 2x - 7 \) is not zero at this \( x \)-value, a vertical asymptote exists.

• At \( x = -\frac{2}{3} \), the line becomes a vertical asymptote.

This means as \( x \) approaches \( -\frac{2}{3} \), the function value \( f(x) \) will go towards infinity or negative infinity, making the graph never cross this line.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) goes towards positive or negative infinity. To find these for rational functions, compare the degrees (highest powers) of the numerator and denominator.

In \( f(x) = \frac{2x - 7}{3x + 2} \), both the numerator and denominator are first-degree polynomials (degree 1). When the degrees are equal:

• The horizontal asymptote is the ratio of the leading coefficients.
• The leading coefficient of the numerator is 2 and for the denominator, it's 3.
• So, the horizontal asymptote is \( y = \frac{2}{3} \).

This line shows where the function levels off as \( x \) becomes very large or small. The function will get closer and closer to \( y = \frac{2}{3} \) but will not actually reach it.