Question

Find the domain of the function and identify any horizontal and vertical asymptotes. $$f(x)=\frac{1-5 x}{1+2 x}$$

Short answer

The domain of the function is all real numbers except \(-1/2\). The function has a vertical asymptote at \(x = -1/2\) and a horizontal asymptote at \(y = 0\).

Step-by-step solution

Find the Domain

Set the denominator of the function equal to zero and solve for \(x\). That is, solve the equation \(1 + 2x = 0\). The solution will be the excluded value from the domain of the function since it would force the denominator to be zero.

Find the Vertical Asymptote

The solution from step 1 will give the vertical asymptote of the function. In other words, vertical asymptote will be the line \(x\) equals to the value from step 1.

Identify Horizontal Asymptote

To find the horizontal asymptote of a rational function, take the degrees of the polynomials in the numerator and denominator into account. If the degree of the denominator is larger than the numerator, the x-axis (y = 0) will be the horizontal asymptote.

Key concepts

Domain of a Function

Understanding the domain of a function is crucial in mathematics, as it sets the stage for where the function actually exists and can be properly evaluated. In simple terms, the domain is the set of all possible input values (often represented by the variable x) for which the function is defined. To find a function's domain, we need to identify the input values for which the function's output remains real and finite.

For the function presented, \( f(x)=\frac{1-5x}{1+2x} \), the domain consists of all real numbers except for the value that makes the denominator zero. This is because division by zero is undefined in mathematics. In this case, setting the denominator \( 1+2x = 0 \) leads to \( x = -\frac{1}{2} \), which is the excluded value from the domain. Hence, the domain of \( f(x) \) is all real numbers except \( x = -\frac{1}{2} \).

When crafting an educational platform's content, it's helpful to encourage students to visualize this concept on a number line or graph, where the excluded value represents a point where the function does not exist.

Horizontal Asymptotes

Understanding Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches but never quite reaches as the input values become very large or very small (as \(x\) approaches infinity or negative infinity). The horizontal asymptote represents a limit to the value that the output of the function (often represented by the variable \(y\)) can approach but not exceed.

For the given function, \(f(x)=\frac{1-5x}{1+2x}\), we determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since the degrees are the same (both are first-degree polynomials), the horizontal asymptote will be the ratio of the leading coefficients. However, in cases where the degree of the numerator is less than that of the denominator, the horizontal asymptote is the x-axis, or \(y=0\).

Allowing students to practice this by comparing degrees and leading coefficients of various rational functions can help reinforce the concept.

Vertical Asymptotes

Identifying Vertical Asymptotes

Vertical asymptotes, as opposed to horizontal ones, occur in locations on a graph where the function heads towards positive or negative infinity as the input value approaches a certain point from both sides. This typically happens when the denominator of a rational function approaches zero, which, as mentioned earlier, is undefined. The vertical asymptote is, in essence, an invisible line that the function approaches infinitely but never crosses or touches.

In the exercise, after setting the denominator to zero and solving for \(x\), we found the excluded value to be \(x=-\frac{1}{2}\). This is where the function is undefined and therefore, represents the vertical asymptote, which in this case is the line \(x=-\frac{1}{2}\).

By guiding students through the process of setting denominators to zero and solving for \(x\), and reinforcing the idea that these values are not included in the domain, we can help them visualize and understand where vertical asymptotes occur on the graph of a function.