Notes on Even and Odd Functions; Parity and Piecewise Example

Even Functions

Definition: A function is even if it is symmetric about the y-axis, meaning for all x in the domain,
f(-x)=f(x).
Graphical intuition: For every point (x, f(x)) there is a corresponding (-x, f(-x)) with the same y-value.
Common examples: f(x)=x^2, f(x)=\cos x (both symmetric about the y-axis).
Practical check: If you mirror the graph across the y-axis, you obtain the same graph.

Definition: A function is odd if it is symmetric about the origin, meaning for all x in the domain,
f(-x)=-f(x).
Graphical intuition: For every point (x, f(x)) there is a corresponding (-x, -f(x)) reflected through the origin.
Common examples: f(x)=\sin x, f(x)=\tan x (both have origin symmetry).
Practical check: If you rotate the graph 180 degrees about the origin, you obtain the same graph.

Step 1: Evaluate f(-x) in terms of x.
Step 2: Compare with f(x) and -f(x):
- If f(-x)=f(x) for all x in the domain, the function is even.
- If f(-x)=-f(x) for all x in the domain, the function is odd.
- If neither condition holds for all x, the function is neither even nor odd.
Important caveat:
- You must check for all x in the domain, not just a few sample points. Parity is a global property.
Domain note: If the domain is not symmetric about 0, parity concepts are limited or may not apply as stated.

Example 1: Let f(x)=x^2. Then f(-x)=(-x)^2=x^2=f(x). Hence even.
Example 2: Let g(x)=x^3. Then g(-x)=(-x)^3=-x^3=-g(x). Hence odd.
Example 3: Let h(x)=x^2+x. Then h(-x)=(-x)^2+(-x)=x^2-x≠h(x) and ≠-h(x) in general, so neither.

Description of the function from the transcript:
- From 0 to 2: a line with slope 30, described as f(t)=30t.
- From 2 to 4: a horizontal line with constant value 60, described as f(t)=60.
Piecewise definition (consistent interpretation):

f(t)=\begin{cases}
30t, & 0\le t\le 2,\
60, & 2< t \le 4.
\end{cases}
Key observations:
- At t=2, both pieces yield 60, so the function is continuous at t=2.
- First segment slope (rise over run):
  m_1=\frac{f(2)-f(0)}{2-0}=\frac{60-0}{2}=30.
- Second segment slope is zero (horizontal line): m_2=0.
- Derivative (where defined):
  
  f'(t)=\begin{cases}
  30, & 0<t<2,\
  0, & 2<t<4,\
  \text{undefined}, & t=2
  \end{cases}
Graphical interpretation:
- A straight line from (0,0) to (2,60) with slope 30.
- A horizontal line at y=60 from t just above 2 to 4.

Symmetry and parity relate to integral properties:
- If f is even, then
  \int{-a}^{a} f(x) \, dx = 2\int{0}^{a} f(x) \, dx.
- If f is odd, then
  \int_{-a}^{a} f(x) \, dx = 0.
Parity concepts underpin Fourier series decomposition:
- Even functions contribute only cosine terms (cosine is even), while odd functions contribute sine terms (sine is odd).
Real-world relevance:
- Symmetry appears in physics (mirror symmetry), engineering (signal processing), and economics (even/odd price functions in certain models). Parity can simplify analyses and computations.

Remember the two defining equations:
- Even: f(-x)=f(x)
- Odd: f(-x)=-f(x)
Always verify on the entire domain, not just a handful of x values.
For piecewise functions, check continuity at breakpoints and note slopes on each interval.
If a piecewise function has a constant segment, its derivative on that interval is zero, and the transition point may be a corner (where the derivative is undefined).

Determine whether the following functions are even, odd, or neither:
- a) f(x)=x^2\cos x
- b) g(x)=\sin x + x^2
Given a piecewise function with a linear segment followed by a constant segment, compute the slope of each part and the derivative on each interval.
Prove that if a function is even, the integral from -a to a equals twice the integral from 0 to a.