Question

Find the domain of the function and identify any horizontal and vertical asymptotes. $$f(x)=\frac{x-7}{5-x}$$

Short answer

The domain of the function is \((-\infty, 5) \cup (5, \infty)\). The function has a vertical asymptote at \(x=5\), and a horizontal asymptote at \(y=-1\).

Step-by-step solution

Find the domain of the function

The domain of the function is all real numbers except those that make the denominator equal to zero. Since the function is \(f(x)=\frac{x-7}{5-x}\), set the denominator equal to zero and solve for 'x':\(5-x = 0 \)\(x = 5\)Therefore, the domain of the function is all real numbers except 5, which can be written in interval notation as \((-\infty, 5) \cup (5, \infty)\).

Identify the vertical asymptote

The vertical asymptote is found by setting the denominator equal to zero, and solving for 'x'. As calculated in step 1, the vertical asymptote is at \(x=5\).

Identify the horizontal asymptote

If the degree of the denominator is greater than the degree of the numerator, the x-axis (y=0) is a horizontal asymptote. Here, the degrees of the numerator and the denominator are the same, so to find the horizontal asymptote, we divide the leading coefficients 1 and -1. So, the horizontal asymptote is \(y = \frac{-1}{1} = -1\).

Key concepts

Domain of a Function

The domain of a rational function, like all other functions, is the set of all possible values of the independent variable (in this case, 'x') for which the function is defined. For rational functions, these are typically all real numbers, except for the values that would make the denominator zero.
When a denominator becomes zero, the function is undefined because division by zero is not allowed in mathematics.
To find the domain, determine which x-values would set the denominator to zero and exclude them.

• For example, in the function \(f(x) = \frac{x-7}{5-x}\), set the denominator equal to zero: \(5-x = 0\).
• Solving this equation gives \(x = 5\), which must be excluded from the domain.

Thus, the domain of this function includes all real numbers except 5, or \(( -\infty, 5) \cup (5, \infty)\) in interval notation.
Recognizing these excluded points is crucial since they often correspond to vertical asymptotes in the function.

Vertical Asymptotes

Vertical asymptotes in rational functions occur at values of 'x' that cause the denominator to be zero (leading to division by zero), while the numerator is not zero at that point.
In simpler terms, a vertical asymptote is a vertical line near which the function grows very large positively or negatively, typically becoming infinite.

• For the function \(f(x) = \frac{x-7}{5-x}\), as seen in the domain calculation, setting the denominator \(5-x\) to zero gives us \(x = 5\).
• This is where the vertical asymptote occurs, as the function cannot exist at this point.

These asymptotes dictate the end behavior of rational functions around specific 'x' values.
It's important to remember that a function will not cross its vertical asymptotes because they represent undefined points for the function.

Horizontal Asymptotes

Horizontal asymptotes in a rational function describe the behavior of the function as 'x' approaches positive or negative infinity.
Essentially, they indicate the value that the function's output will get closer and closer to, but never reach, as 'x' becomes very large or very small.
To determine horizontal asymptotes, compare the degrees of the polynomials in the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y=0\).
• If both have the same degree, as in our example, divide the leading coefficients of the numerator by the denominator to find the asymptote.
• In our case \(f(x) = \frac{x-7}{5-x}\), the degrees are the same, and dividing the coefficients 1 and -1 gives \(y = -1\).

This line \(y = -1\) is the horizontal asymptote that the function approaches as the values of 'x' increase or decrease indefinitely.
While functions can cross horizontal asymptotes, they tend not to very often, especially far out on either end of the x-axis. This behavior emphasizes the "flattening out" effect that horizontal asymptotes often predict.