Reciprocal of Linear and Quadratic Functions
The reciprocal of a function \(f(x)\) is given by \(g(x) = \frac{1}{f(x)}\).
Reciprocal of a Linear Function
Consider a linear function \(f(x) = mx + b\). The reciprocal is \(g(x) = \frac{1}{(mx + b)}\).
Graphing:
- If \(f(x) = 0\), then g(x) is undefined, resulting in a vertical asymptote.
- If \(f(x)\) is positive, then g(x) is also positive.
- If \(f(x)\) is negative, then g(x) is also negative.
Example
Let’s graph the reciprocal of the function \(f(x) = 2x + 1\).
- Identify the zeros of \(f(x): x = \frac{-1}{2}\) (vertical asymptote).
- Determine the sign of \(f(x)\): negative for \(x < \frac{-1}{2}\), positive for \(x > \frac{-1}{2}\).
- Sketch the graph of g(x) using this information.
Reciprocal of a Quadratic Function
Now consider a quadratic function \(f(x) = ax^2 + bx + c\). The reciprocal is \(g(x) = \frac{1}{(ax^2 + bx + c)}\).
Graphing:
- If \(f(x) = 0\), then \(g(x)\) is undefined, resulting in a vertical asymptote.
- If \(f(x)\) is positive, then \(g(x)\) is also positive.
- If \(f(x)\) is negative, then \(g(x)\) is also negative.
Example
Let’s graph the reciprocal of the function \(f(x) = x^2 - 1\).
- Identify the zeros of \(f(x): x = -1\) and \(x = 1\) (vertical asymptotes).
- Determine the sign of \(f(x)\): negative for \(-1 < x < 1\), positive for \(x < -1\) or \(x > 1\).
- Sketch the graph of \(g(x)\) using this information.
What are Holes and Asymptotes? How to Find Them?
A hole in a function is a point where the function is undefined. In a reciprocal function, holes occur where the denominator is zero.
An asymptote is a line that the graph of the function approaches but never touches. In a reciprocal function, vertical asymptotes occur where the function is undefined, and the horizontal asymptote is usually the \(x-axis\) (\(y=0\)), unless the function is a rational function with the degree of the numerator equal to or greater than the degree of the denominator.
Types of Asymptotes and How to Find Each?
There are three types of asymptotes: vertical, horizontal, and oblique.
Vertical Asymptotes:
These are vertical lines (of the form \(x = a\)) that the graph approaches but never crosses. To find vertical asymptotes, set the denominator equal to zero and solve for x.
Horizontal Asymptotes:
a) When the degree of the numerator is greater than the degree of the denominator: In this case, the function does NOT have a horizontal asymptote. Instead, it may have an oblique (slant) asymptote, which occurs when the degree of the numerator is exactly one more than the degree of the denominator.
b) When the degree of the numerator is less than the degree of the denominator: If the degree of the numerator is less than the degree of the denominator, the x-axis (\(y = 0\)) is the horizontal asymptote. This is because as \(x\) approaches infinity, the value of the function approaches 0.
c) When the degree of the numerator is equal to the degree of the denominator: If the degrees are equal, the horizontal asymptote is found by dividing the coefficients of the terms with the highest degree. For example, for the function \(f(x)=\frac{(3x^2+2x+1)}{(2x^2-x+3)}\), the horizontal asymptote is \(y = \frac{3}{2}\), which is the ratio of the coefficients of the \(x^2\) terms.
Oblique Asymptote:
An oblique asymptote, also known as a slant asymptote, occurs when the degree of the numerator of a rational function is one more than the degree of the denominator. To find an oblique asymptote:
- Perform polynomial long division on the function.
- The quotient (ignoring the remainder) is the equation of the oblique asymptote.
For example, consider the function \(f(x)=\frac{(2x^2+3x-2)}{(x-1)}\). Performing long division gives us \(2x + 5\) as the quotient. So, the oblique asymptote is the line \(y = 2x + 5\). As x approaches infinity or negative infinity, the graph of the function gets closer and closer to the line \(y = 2x + 5\), but never actually reaches it. This is the concept of an oblique asymptote.
Graphing Reciprocal Functions
Graphing reciprocal functions involves plotting points, analyzing the behavior near asymptotes, and identifying special features like holes. By following a step-by-step process, we can create an accurate representation of reciprocal functions on a graph.
Here are the steps to graph a reciprocal function:
- Determine the domain and range of the function.
- Identify any holes or asymptotes present.
- Plot points using a table of values.
- Mark the location of vertical and horizontal asymptotes.
- Connect the plotted points smoothly, considering the behavior near asymptotes and any holes present.
Example
Graph the reciprocal function \(f(x) = \frac{1}{x}\).
Solution: Following the steps mentioned above, we obtain the graph as shown below:
Conclusion
The reciprocal of a function provides interesting insights into the behavior of the original function. By studying the reciprocal, we can better understand the function’s zeros and intervals of positivity and negativity. As always, practice is key to mastering these concepts. Happy studying!
Practice Problems
- Graph the reciprocal of the function \(f(x) = 3x - 2\)
- Graph the reciprocal of the function \(f(x) = -x^2 + 4\).
- For the function \(f(x) = x^2 - 4x + 4\), find the x-values where the reciprocal function is undefined.
- For the function \(f(x) = 2x + 3\), find the x-values where the reciprocal function is positive.