Question

A function \(f\) is given on an interval of length \(L .\) In each case sketch the graphs of the even and odd extensions of \(f\) of period \(2 L .\) $$ f(x)=4-x^{2}, \quad 0

Short answer

The even extension of the given function is \(f_{even}(x) = 4 - x^2\), and the odd extension is \(f_{odd}(x) = 0\).

Step-by-step solution

Let's first define the even and odd extensions

The even extension of a function is defined as \(f_{even}(x) = \frac{1}{2}(f(x) + f(-x))\) and the odd extension as \(f_{odd}(x) = \frac{1}{2}(f(x) - f(-x))\). Let's find these for our function, \(f(x) = 4 - x^2\).

Find the even and odd extensions of \(f(x) = 4 - x^2\)

Let's apply our definitions of even and odd extensions to \(f(x)\): For the even extension: $$ f_{even}(x) = \frac{1}{2}((4 - x^2) + (4 - (-x)^2)) = \frac{1}{2}((4 - x^2) + (4 - x^2)) = 4 - x^2 $$ For the odd extension: $$ f_{odd}(x) = \frac{1}{2}((4 - x^2) - (4 - (-x)^2)) = \frac{1}{2}((4 - x^2) - (4 - x^2)) = 0 $$ So we have that our even extension function is \(f_{even}(x) = 4 - x^2\), which is the original function, and our odd extension function is \(f_{odd}(x) = 0\).

Sketch the even extension function

Now let's sketch the even extension function, \(f_{even}(x) = 4 - x^2\). To do this, we'll consider the period of the function, which is \(2L\). Since we know the function is defined on the interval \((0, 1)\), we'll use this information to sketch the function. For the even extension, the given function's graph is simply reflected about the y-axis. Thus, it will look like a parabola centered at \(x=0\) and opening downward. Within the interval of \((0, 1)\), the function maintains \(f(x) = 4 - x^2\). For other intervals, the function repeats with the same shape periodically and symmetrically.

Sketch the odd extension function

Now let's sketch the odd extension function, \(f_{odd}(x) = 0\). Our period is \(2L\), and since the function is always 0, the graph will be a straight line at the x-axis. In conclusion, we have determined the even extension \(f(x)= 4-x^{2}\) that remains the same as the original function and the odd extension \(f(x)=0\) which is a straight line at the x-axis. We can now sketch both extensions using the identified interval and period.

Key concepts

Periodicity

Periodicity is a fundamental concept in understanding periodic functions. It refers to the property that a function repeats its values in regular intervals or periods.
For a function defined on an interval of length \(L\), the concept of periodicity implies that the function will exhibit the same behavior every \(2L\) units when sketched in a graph.
In our example, the function \(f(x) = 4 - x^2\), defined for \(0Understanding periodicity is crucial when working with extensions, as it determines the visual repetition of the function's pattern.

• Even Extensions: Mirror the function's graph over the y-axis.
• Odd Extensions: Reflect across the origin, resulting in a different repetitive pattern.

The knowledge of periodicity assists in predicting how the graph will behave beyond its initial definition boundaries.

Parabolic Function

A parabolic function is represented in the form \(f(x) = ax^2 + bx + c\). Its graph forms a shape known as a parabola.
The given function \(f(x) = 4 - x^2\) is a downward-opening parabola because the coefficient of \(x^2\) is negative (-1).
The characteristics of this parabolic function include:

• Vertex: The peak point of the parabola. For \(f(x) = 4 - x^2\), the vertex is at \((0, 4)\) when considered over its complete domain.
• Axis of symmetry: A vertical line that runs through the vertex, \(x = 0\), symmetrically dividing the parabola.
• Opening: As the function has \(-x^2\), it means the parabola opens downwards.

Understanding these features is vital when sketching, as it helps align the properties with the description of either the original, even, or odd extensions.

Graph Sketching

Graph sketching involves visually representing functions to understand their behavior over a given domain and any extended periods.
For the function \(f(x) = 4 - x^2\):
- **Even Extension:** This graph will look precisely like the original function due to the symmetry about the y-axis. In a typical even extension, \(f(x)\) replicates itself across the y-axis, thereby creating a single, repeated parabola in each \(2L\) domain interval. Visualizing this helps us see its recurrence pattern.- **Odd Extension:** For this, the graph of the function would be zero across the x-axis because it does not maintain any definable curve in its odd form. This essentially means you'll draw a flat line at \(y=0\) within each interval of \(2L\).
Key considerations for graph sketching:

• Ensure the function matches its defined properties, such as the vertex position and parabola direction (up or down).
• Recognize the full periodic extension, sketching regularly repeated intervals as dictated by the periodicity.

By sketching the graph, students can visually comprehend how the function behaves, both originally and in extended periods, highlighting the underlying symmetry and periodicity.