Question

Write a function M-file to calculate \(f(z)=\sqrt{|z|}\). Use it to graph \(f\) over the interval \((-3,2)\).

Short answer

Define a MATLAB function for \( f(z) = \sqrt{|z|} \), use `arrayfun` and `plot` to graph it over \((-3,2)\).

Step-by-step solution

Define the function

We start by creating a MATLAB function to calculate \( f(z) = \sqrt{|z|} \). Open a new script in MATLAB and define the function as follows:```matlabfunction y = f(z) y = sqrt(abs(z));end```This function takes an input \( z \) and calculates the square root of the absolute value of \( z \).

Create a script for plotting

We now need a script to call this function over the specified interval and plot the results. Create a new script, and add the following lines:```matlabz_values = linspace(-3, 2, 100);f_values = arrayfun(@f, z_values);plot(z_values, f_values);xlabel('z');ylabel('f(z) = \sqrt{|z|}');title('Plot of f(z) over the interval (-3, 2)');````linspace` generates 100 points between -3 and 2. `arrayfun` applies the function \( f(z) \) to each point in `z_values`. The plot function visualizes the output.

Execute the script and interpret the plot

After saving the script, run it in MATLAB. This will generate and display a plot of \( f(z) \) over the interval from -3 to 2. In the plot, observe that since we are taking the absolute value inside the square root, the function will not produce complex numbers for negative inputs, but will instead reflect the symmetry along the y-axis for the magnitude of negative inputs.

Key concepts

Function Definition

A function in MATLAB is defined as a script that performs specific operations. It takes input values and returns outputs. In MATLAB, a function definition starts with the keyword "function," which describes its purpose and inputs. For example, the function for calculating \( f(z) = \sqrt{|z|} \) is defined with the line:
function y = f(z)
This line states that the function name is `f`, it takes an input `z`, and returns an output `y`. The operations inside the function are enclosed between function and "end" keyword. The code y = sqrt(abs(z)); calculates the square root of the absolute value of `z`.

• The function allows reuse, meaning you can call it multiple times with different input values.
• Having a separate function makes your code organized and easy to understand.

Using functions in MATLAB automates calculations and can handle variable input sizes effectively. This makes MATLAB a powerful tool for mathematical computations.

Plotting Graphs

Graphing is essential in MATLAB to analyze data visually. To plot a graph, you need data points and a plotting command. In this exercise, the `plot` function is used. This function generates a graph of \( f(z) = \sqrt{|z|} \) over the range \((-3, 2)\). Using the code:
plot(z_values, f_values),
the x-axis is composed of `z_values` and the y-axis of `f_values`. The plot function also supports additional parameters to make the graph informative.

• The `xlabel` and `ylabel` functions label each axis, clarifying what each axis represents.
• The `title` function explains what the graph depicts.

Graphing is an invaluable part of MATLAB, offering a dynamic view of data relationships. It helps in identifying trends and features that may not be apparent from raw data alone. The combination of labeling and titles helps make a polished and professional-looking graph.

Abs Function

In MATLAB, the `abs` function is used to calculate the absolute value of a numeric input. For example, if you provide a negative number to `abs`, it will return the positive counterpart of that number.
This function is crucial in many mathematical calculations where the magnitude of a number is needed regardless of its sign. For instance, in the function \( f(z) = \sqrt{|z|} \), the `abs` function ensures the argument to the square root is always non-negative, preventing the generation of complex numbers.

• Syntax: `abs(x)` where `x` is a scalar, vector, or matrix.
• It is straightforward and guarantees the outcome is always a non-negative number.

The `abs` function simplifies calculations by eliminating potential negative value issues, ensuring precise results. This makes it a valuable tool in programming and data analysis within MATLAB.

Linspace Function

The `linspace` function in MATLAB is a practical tool for generating vectors with evenly spaced elements. This function is vital when you need to create a sequence of numbers over a specific interval, such as \((-3, 2)\).
The syntax: `linspace(start, end, num_points)` where `start` and `end` define the interval range, and `num_points` sets how many points will be generated.

• `linspace(-3, 2, 100)` creates 100 evenly spaced values between -3 and 2.
• This is particularly useful in plotting, where you need consistent intervals between points for smooth curves.

Using `linspace` ensures precision in defining domains for plotting functions or simulations, offering a controlled approach to graph visualizations in MATLAB. It simplifies the creation of grids, ensuring your analysis covers the entire specified range adequately.