Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ \phi(z)=\frac{2 z+1}{z-1} $$

Short answer

The function is neither even nor odd.

Step-by-step solution

Understand the Definitions

A function is _even_ if for every value of \(z\), \( \phi(-z) = \phi(z) \). It is _odd_ if \( \phi(-z) = -\phi(z) \) for every \(z\). If neither condition is met, the function is _neither even nor odd_.

Substitute and Simplify \(\phi(-z)\)

Substitute \(-z\) into the function: \[\phi(-z) = \frac{2(-z) + 1}{-z - 1} = \frac{-2z + 1}{-z - 1}. \] Now simplify:\[\phi(-z) = \frac{-2z + 1}{-z - 1} = \frac{-(2z - 1)}{-(z + 1)} = \frac{2z - 1}{z + 1}.\]

Compare \(\phi(z)\) and \(\phi(-z)\)

Original function: \(\phi(z) = \frac{2z + 1}{z - 1}\). Substituted function: \(\phi(-z) = \frac{2z - 1}{z + 1}\). Since \(\phi(-z) eq \phi(z)\) and \(\phi(-z) eq -\phi(z)\), the function is neither even nor odd.

Sketch the Graph of \(\phi(z)\)

To sketch \(\phi(z) = \frac{2z + 1}{z - 1}\), note the vertical asymptote at \(z = 1\) where the denominator is zero. The horizontal asymptote is \(y = 2\) as \(z\) approaches infinity. Plot a few key points such as at \(z = 0\) (where \(\phi(0) = -1\)), and sketch the behavior near the asymptotes and intercepts.

Key concepts

Function Symmetry

Symmetry in functions refers to the behavior of the function with respect to the y-axis and the origin. To determine if a function is even or odd, we look for symmetry:

• A function is even if it is symmetrical with the y-axis. This means that if you reflect the graph across the y-axis, you will get the same graph. Mathematically, this means for the function \(\phi(z)\), we have \(\phi(-z) = \phi(z)\). Examples of even functions include \(z^2\) and \(\cos(z)\).


• Odd functions are symmetrical with respect to the origin. This means that when you rotate the graph 180 degrees about the origin, the graph remains unchanged. For a function \(\phi(z)\) to be odd, \(\phi(-z) = -\phi(z)\) must hold. Common odd functions are \(z^3\) and \(\sin(z)\).


• If neither condition is satisfied, the function is neither even nor odd, like the function \(\phi(z) = \frac{2z + 1}{z - 1}\) in the exercise.

Graph Sketching

Sketching the graph of a function helps us to visually analyze the function's behavior and key characteristics. To sketch the graph of \(\phi(z) = \frac{2z + 1}{z - 1}\):

• Identify key features like intercepts. For example, the y-intercept occurs when \(z = 0\), leading to \(\phi(0) = -1\).


• Check where the graph crosses the x-axis by setting \(\phi(z) = 0\), which provides insights about the zeros of the function.


• Observe the behavior near the asymptotes. In this function, both vertical and horizontal asymptotes provide crucial information about the general shape of the graph.


• Finally, plot a few additional points if necessary and connect them to illustrate the function's continuous flow.

Vertical Asymptotes

Vertical asymptotes represent locations where the function value tends to infinity. It indicates the x-values for which the function does not exist.

• For \(\phi(z) = \frac{2z + 1}{z - 1}\), set the denominator equal to zero to find vertical asymptotes: \(z - 1 = 0\), giving \(z = 1\).


• The graph approaches these lines but never actually touches or crosses them.


• Understanding vertical asymptotes allows us to predict where the function will seem to "shoot up" or "shoot down" towards infinity, providing important insights when sketching.

Horizontal Asymptotes

Horizontal asymptotes help describe the end behavior of the function as \(z\) tends towards positive or negative infinity.

• For the function \(\phi(z) = \frac{2z + 1}{z - 1}\), as \(z\) becomes very large or very small, the impact of the constants becomes negligible, stabilizing the function's value.


• To find the horizontal asymptote, compare the degrees of the numerator and denominator. Here, both have degree 1, so divide the leading coefficients: \(\lim_{{z \to \infty}} \frac{2z + 1}{z - 1} = 2\). Therefore, the horizontal asymptote is \(y = 2\).


• This helps predict how the graph behaves as it extends far from the origin, either towards +∞ or -∞, providing valuable insights during graph sketching.