2.16 Symmetries of f(x) - More Transformations (HL)

1. Even and Odd Functions

Even Functions: Defined mathematically when $f(-x) = f(x)$. The function is perfectly symmetrical across the geometric y-axis. The base parameter example is $f(x) = x^2$.
Odd Functions: Defined mathematically when $f(-x) = -f(x)$. The function translates symmetrically around the origin point $(0,0)$. The base parameter example is $f(x) = x^3$.

EXAMPLE 1

Algebraically investigate whether designated functions display even or odd characteristics.

(a) $f(x) = x^4 + |x|$
Evaluate input mapping: $f(-x) = (-x)^4 + |-x| = x^4 + |x| = f(x)$. This confirms an even function.
(b) $g(x) = x - x^3$
Evaluate input mapping: $g(-x) = (-x) - (-x)^3 = -x + x^3 = -(x - x^3) = -g(x)$. This confirms an odd function.
(c) $h(x) = x + x^2$
Evaluate input mapping: $h(-x) = (-x) + (-x)^2 = -x + x^2$. The output equals neither $h(x)$ nor $-h(x)$. The function acts uniquely neither even nor odd.

2. Absolute Value Transformations

Applying the absolute modulus alters distinct geometric visual properties on any foundational curve $f(x)$.

The Transformation $|f(x)|$:
This strictly affects the operational outputs. It securely preserves any naturally positive graphical elements, while immediately reflecting all negative geometric intervals aggressively across the boundary x-axis (because negative values logically flip to $-f(x)$).
The Transformation $f(|x|)$:
This explicitly evaluates incoming domain bounds. It fundamentally ignores all existing curve geometry where $x < 0$. It then duplicates the entire existing positive graphical space ($x \ge 0$) as a clean mirrored reflection stretching across the boundary y-axis (because any negative input identically calculates the positive counterpart output).

3. The Reciprocal Function $1/f(x)$

Inverting the outputs establishes strict graphical rules guiding asymptotes, roots, and peak thresholds.

To sketch the graph of the reciprocal $g(x) = \frac{1}{f(x)}$, the following mechanical rules are utilized:

  1. Every root intersection ($x$-intercept) mapping functionally to zero mathematically crashes the reciprocal, converting strictly to a Vertical Asymptote. Correspondingly, previous Vertical Asymptotes flip entirely into root zero-crossings.
  2. A given Horizontal Asymptote tracing $y=a$ functionally transforms entirely to $y = 1/a$. If $y=0$ defines a base Horizontal Asymptote, the new reciprocal shoots unbounded toward $+\infty$ or $-\infty$.
  3. Identified critical turning points scale fractionally: A local coordinate maximum mapping $(x, y)$ collapses inversely into a local minimal depression tracking $(x, 1/y)$. Any minimum symmetrically acts vice-versa.
  4. Positive output sectors reliably remain structurally positive. Negative output sectors rigorously remain negative.
  5. Where $f(x)$ exhibits upward increasing behavior, $1/f(x)$ inversely dictates decreasing graphical paths.

4. The Squared Transformation $[f(x)]^2$

Squaring an entire function alters geometry uniquely and non-linearly across boundary threshold intervals.

  • Any mathematical points established precisely on the $y=1$ or $y=0$ bounds strictly remain locked identical on the curve.
  • Values tracking strictly at $y=-1$ universally flip structurally upwards translating perfectly to $y=1$.
  • Coordinate geometry traversing higher than $y=1$ accelerates exponentially upwards (stretching visually: $2$ scales aggressively to $4$).
  • Sub-unit coordinate elements traversing mathematically between $0 < y < 1$ depress further downwards towards the root floor (shrinking visually: $0.5$ evaluates deeply to $0.25$).
  • All sub-zero negative intervals shift structurally to the top-half grid, maintaining similar scale-stretching factors as derived positive segments.