Question

Find the domain of the function and identify any horizontal and vertical asymptotes. $$f(x)=\frac{3 x}{x+1}$$

Short answer

The domain of the function \(f(x) = \frac{3x}{x+1}\) is \(x \neq -1\). The function has a horizontal asymptote at \(y = 3\), and a vertical asymptote at \(x = -1\).

Step-by-step solution

Identify the Domain

The domain of \( f(x) = \frac{3x}{x+1} \) are all real numbers except where the denominator is 0. Thus, to find the domain, solve the equation \(x+1 = 0\) for x. This gives \(x = -1\). So, the domain of this function is \(x \neq -1\).

Find the Horizontal Asymptote

Horizontal asymptotes can be identified by taking the limit as x approaches positive and negative infinity. By analyzing the function, it can be seen that as x approaches infinity, the function approximates 3 and as x approaches negative infinity, the function also approximates 3. Hence the function has a horizontal asymptote at \(y = 3\).

Locate the Vertical Asymptote

Vertical asymptotes can be identified by finding the values of x where the function is undefined. As found in step 1, the denominator can not be equal to 0, so the vertical asymptote will be at the point \(x = -1\).

Key concepts

Finding Domain in Algebra

In algebra, finding the domain of a function is a fundamental skill. The domain encompasses all possible input values (usually represented by the variable x) for which the function is defined. To find the domain, we need to identify any restrictions on the input values. Common restrictions include divisions by zero and negative square roots, which are undefined in real numbers.

For the function presented, \(f(x) = \frac{3x}{x+1}\), the domain is all real numbers, except those that make the denominator zero. In this case, set the denominator equal to zero and solve for x: \(x + 1 = 0\), giving us \(x = -1\). Therefore, the domain of this function is all real numbers except for \(x = -1\). By highlighting the excluded value, it becomes clear to students that \(x = -1\) is the 'hole' in the function where it is not defined.

Horizontal Asymptote Determination

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as the input value heads towards positive or negative infinity. To determine the horizontal asymptote, we analyze the behavior of the function as \(x\) approaches infinity and negative infinity.

The function \(f(x) = \frac{3x}{x+1}\) will have its long-term behavior mirrored in the ratio of the leading coefficients of the numerator and denominator when the degrees are the same. Since both the numerator and denominator are of first degree, the coefficients are 3 and 1 respectively. Therefore, as \(x\) approaches infinity in either direction, the value of \(f(x)\) approaches the ratio of these coefficients, which is 3. This makes the horizontal asymptote \(y = 3\).

Vertical Asymptote Identification

Vertical asymptotes occur in the graph of a function at the input values where the function becomes unbounded or undefined. To identify vertical asymptotes, we must look for places where the denominator of a function equals zero, but only if the numerator does not also equal zero at those points.

Looking at the given function, \(f(x) = \frac{3x}{x+1}\), we already know from the domain determination that \(x = -1\) is where the function is undefined. This is because the denominator becomes zero and the function value tends to infinity. Hence, we identify \(x = -1\) as a vertical asymptote. This concept reinforces the understanding of why a particular value is excluded from the domain of the function.

Real Numbers Exclusion for Function Domain

Excluding real numbers from the domain of a function is essential when those numbers result in undefined or imaginary terms, which are not acceptable in real-valued functions. Such exclusions typically result from zeros in denominators or even roots of negative numbers.

When we speak of the domain of \(f(x) = \frac{3x}{x+1}\), we exclude \(x = -1\) because it causes division by zero. This action is critical for maintaining the function within the realm of real numbers, ensuring that every value within the domain results in a real number output. This step is vital for students to understand the boundaries of a function's domain in relation to the set of real numbers.