Question

Construct a function with a vertical asymptote at \(x=5\) and a horizontal asymptote at \(y=5\).

Short answer

The function is \(f(x) = \frac{5x}{x-5}\).

Step-by-step solution

Identify Conditions for Asymptotes

To construct the function, we need it to have a vertical asymptote at \(x=5\) and a horizontal asymptote at \(y=5\). A vertical asymptote occurs when the denominator of a function is zero, and a horizontal asymptote \(y=L\) can be achieved when the limit of the function as \(x\) approaches infinity is \(L\).

Choose a Function Form

A rational function is a good candidate: \(f(x) = \frac{ax+b}{x-c}\). The denominator causes vertical asymptotes, and if \(c=5\), the function will have a vertical asymptote at \(x=5\).

Adjust Numerator for Horizontal Asymptote

We want the function to have a horizontal asymptote at \(y=5\). For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is \(y=\frac{a}{1}\). So set \(a=5\) to get \(y=5\).

Final Function Construction

Based on these conditions, construct the function \(f(x) = \frac{5x+b}{x-5}\). The specific value of \(b\) does not affect the asymptotic behavior, so we can set \(b=0\) for simplicity. Thus, the function is \(f(x) = \frac{5x}{x-5}\).

Key concepts

Vertical Asymptote

A vertical asymptote is a line where the graph of a function tends to infinity. It signifies that the function cannot produce a definite value for a specific input, often because the denominator of a rational function equals zero. For the function \(f(x) = \frac{5x}{x-5}\), there is a vertical asymptote at \(x=5\). This occurs because, when \(x\) approaches 5, the denominator becomes zero, making the function undefined at this point.

It's crucial to recognize that vertical asymptotes indicate where a function's values increase without bound in the positive or negative direction. They inform us about the behavior of a function near specific x-values, helping us understand the function's graph.

• Vertical asymptotes occur at points where the denominator is zero while the numerator is non-zero.
• They indicate a boundary in the graph where the function tends toward positive or negative infinity.
• They are crucial in pinpointing where a function's graph will veer off the visible plane.

Horizontal Asymptote

Horizontal asymptotes tell us the behavior of a function as \(x\) approaches infinity or negative infinity. They give insight into the function's limiting value along the y-axis. For a rational function like \(f(x) = \frac{5x}{x-5}\), if the degrees of the numerator and the denominator are the same, the horizontal asymptote can be found by dividing the leading coefficients. Therefore, \(f(x)\) has a horizontal asymptote at \(y=5\).

This means as \(x\) becomes very large (positive or negative), the value of \(f(x)\) approaches 5. Regardless of how big \(x\) gets, the function will get closer and closer to \(y=5\) but never actually reach this line. It's vital to understand that horizontal asymptotes provide a sort of evening ground in graphs by demonstrating limits for outputs as inputs grow tremendously.

• Horizontal asymptotes describe the end behavior of a function as \(x\) approaches infinity.
• They are especially crucial when dealing with rational functions and their simple interpretation through leading coefficients.
• The presence of a horizontal asymptote shows us the end behavior of the function, showing it grows closer to a specific y-value.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. It is generally represented as \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. Rational functions are significant because they can model a wide range of behaviors and are often used to represent real-world problems.

The presence of both vertical and horizontal asymptotes in rational functions helps provide a clearer picture of their behavior. In the function \(f(x) = \frac{5x}{x-5}\), we relied on rational function properties to easily set asymptote conditions:

• Vertical asymptotes arise from zeroes in the denominator where the function is undefined.
• Horizontal asymptotes are based on the degrees of the top and bottom polynomials. Here, they helped establish a limiting value for output.
• They help us understand key points of behavior that are critical in sketching the function's graph.

Understanding how to work with rational functions and their asymptotes is fundamental in mathematics, providing substantial insight into complex graph behaviors.