Question

Graph the function. Describe the domain. $$y=\frac{9 x-6}{x-1}$$

Short answer

The domain of the function \(y=\frac{9x-6}{x-1}\) is all real numbers except 1. The function has its x-intercept at \((\frac{2}{3}, 0)\) and y-intercept at \((0, -6)\). It has lines \(x=1\) and \(y=9\) as its vertical and horizontal asymptotes, respectively. The function is graphed accordingly.

Step-by-step solution

Domain Determination

The function is undefined when the denominator of the fraction is equal to zero, i.e., \(x-1=0\). Solving for \(x\) results in \(x=1\). Therefore, the domain of the function is all real numbers except 1.

Finding the X and Y Intercepts

The x-intercept can be found by setting \(y=0\), which results in \(9x-6=0\). Solving for \(x\) gives \(x=\frac{2}{3}\), so the x-intercept is at \((\frac{2}{3}, 0)\). The y-intercept is found by setting \(x=0\), which results in \(y=-6\), so the y-intercept is at \((0, -6)\).

Identifying the Asymptotes

There is a vertical asymptote at \(x=1\) since it's the value that makes the function undefined. The horizontal asymptote can be determined by finding the limit of the function as \(x\) approaches infinity, this gives us \(y=9\). Therefore, the horizontal asymptote is the line \(y=9\).

Graphing the Function

Plot the x and y intercept(s). Draw the vertical and horizontal asymptotes. Then graph the function using these points and lines as guides. Remember that the graph will approach the asymptotes but never cross them.

Key concepts

Domain of a Function

Understanding the domain of a function is crucial when it comes to graphing rational functions. In simpler terms, the domain consists of all the input values, or 'x' values, for which the function is defined.

For the rational function
\[y=\frac{9x-6}{x-1}\]
we must exclude any 'x' values from the domain that cause the denominator to be zero, as division by zero is undefined. In this case, setting the denominator equal to zero,
\[x-1=0\]
shows us that 'x' cannot be 1. Therefore, the domain is all real numbers except for 1. This concept is a fundamental step in graphing and understanding rational functions.

X-Intercepts and Y-Intercepts

The points at which a graph crosses the x-axis are known as x-intercepts, and they are found by setting 'y' to zero and solving for 'x'. For our function,
\[y=\frac{9x-6}{x-1}\]
the x-intercept occurs at
\[x=\frac{2}{3}\]
as that is the solution to
\[9x-6=0\].

The y-intercept is found in a similar manner, by setting 'x' to zero. This gives us the point (0, -6), as when
\[x=0\]
it leads to
\[y=-6\].
Graphing these intercepts helps give shape to the function and aids in understanding its behavior near the axes.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never quite reaches. They represent values outside the function's domain where the function increases or decreases without bound.

In our function, the vertical asymptote is at
\[x=1\]
because as 'x' gets closer to 1, the value of the function grows indefinitely. This occurs because the denominator of our function
\[x-1\]
approaches zero, which makes the value of the whole fraction very large in magnitude. Vertical asymptotes are critical in understanding the behavior of functions and are a key feature when sketching their graphs.

Horizontal Asymptotes

While vertical asymptotes are about the function's behavior in relation to the 'x' values, horizontal asymptotes relate to the behavior as 'x' gets very large or very small—in other words, as
\[x\to\pm\infty\].

For the rational function
\[y=\frac{9x-6}{x-1}\],
the horizontal asymptote is the line
\[y=9\].
It predicts the value that 'y' approaches as 'x' goes to infinity. If the degrees of the polynomial in the numerator and the denominator are the same, the coefficients of the highest degree terms determine the horizontal asymptote. In this case, both the numerator and denominator are linear, so the coefficients of
\[x\]
are what matter, leading to an asymptote of
\[y=9\].
Understanding horizontal asymptotes is important for predicting end behavior of rational functions.