3.2 Function Symmetry, Extrema, and Operations

3.2 Function Symmetry, Extrema, and Operations

1. Symmetry of Functions: Even vs. Odd

Analyzing function symmetry allows precalculus students to simplify graphing, anticipate function behavior, and solve algebraic equations more efficiently.

Even Functions (y-Axis Symmetry)

A function $f$ is even if replacing $x$ with $-x$ leaves the function expression entirely unchanged: $f(-x) = f(x) \quad \text{for all } x \text{ in the domain}$ Geometrically, the graph of an even function is symmetric with respect to the y-axis. Folding the graph along the vertical axis produces a perfectly matching image on both sides.

Odd Functions (Origin Symmetry)

A function $f$ is odd if replacing $x$ with $-x$ yields the exact negation of the original function: $f(-x) = -f(x) \quad \text{for all } x \text{ in the domain}$ Geometrically, the graph of an odd function is symmetric with respect to the origin $(0,0)$. Rotating the graph $180^\circ$ around the origin produces the original graph.

Worked Example 2.2A: Testing Symmetry Algebraically

Classify each function as even, odd, or neither: (a) $f(x) = \frac{x^3}{x^2 + 4}$, (b) $g(x) = 3x^4 - 2x^2 + 7$, (c) $h(x) = x^3 + x^2$.

Analysis for $f(x)$: $f(-x) = \frac{(-x)^3}{(-x)^2 + 4} = \frac{-x^3}{x^2 + 4} = -\left(\frac{x^3}{x^2 + 4}\right) = -f(x)$ Since $f(-x) = -f(x)$, $f(x)$ is odd.

Analysis for $g(x)$: $g(-x) = 3(-x)^4 - 2(-x)^2 + 7 = 3x^4 - 2x^2 + 7 = g(x)$ Since $g(-x) = g(x)$, $g(x)$ is even.

Analysis for $h(x)$: $h(-x) = (-x)^3 + (-x)^2 = -x^3 + x^2$ Compare with $h(x) = x^3 + x^2$ and $-h(x) = -x^3 - x^2$. Since $h(-x)$ matches neither $h(x)$ nor $-h(x)$, $h(x)$ is neither.

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2. Function Intervals and Extrema

Increasing, Decreasing, and Constant Intervals

Let $f$ be a function defined on an interval $I$. For any $x_1, x_2 \in I$ with $x_1 < x_2$:

• $f$ is increasing on $I$ if $f(x_1) < f(x_2)$ (graph rises from left to right).
• $f$ is decreasing on $I$ if $f(x_1) > f(x_2)$ (graph falls from left to right).
• $f$ is constant on $I$ if $f(x_1) = f(x_2)$ (graph is flat horizontal line).

Note: Intervals of increasing/decreasing/constant behavior are always expressed using open interval notation $(a, b)$ of $x$-values.

Local (Relative) and Global (Absolute) Extrema

• Relative Maximum: A value $f(c)$ is a relative maximum if $f(c) \ge f(x)$ for all $x$ in an open interval containing $c$ ("peak of a hill").
• Relative Minimum: A value $f(c)$ is a relative minimum if $f(c) \le f(x)$ for all $x$ in an open interval containing $c$ ("bottom of a valley").
• Absolute Maximum: The absolute highest $y$-value over the entire domain.
• Absolute Minimum: The absolute lowest $y$-value over the entire domain.

Average Rate of Change (AROC)

The average rate of change of $f(x)$ over the interval $[a, b]$ measures the average speed at which the output changes per unit change in input: $\text{AROC} = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$ Geometrically, this value represents the slope of the secant line passing through points $(a, f(a))$ and $(b, f(b))$ on the graph of $f$.

Worked Example 2.2B: Average Rate of Change

Calculate the average rate of change of $f(x) = x^3 - 4x + 2$ on $[1, 3]$.

Step 1: Compute function values at endpoints. $f(1) = (1)^3 - 4(1) + 2 = 1 - 4 + 2 = -1$ $f(3) = (3)^3 - 4(3) + 2 = 27 - 12 + 2 = 17$

Step 2: Apply the difference quotient formula. $\text{AROC} = \frac{f(3) - f(1)}{3 - 1} = \frac{17 - (-1)}{2} = \frac{18}{2} = 9$

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3. Operations on Functions and Algebra of Functions

Given two functions $f(x)$ and $g(x)$, we define four standard arithmetic operations:

1. Sum: $(f + g)(x) = f(x) + g(x)$
2. Difference: $(f - g)(x) = f(x) - g(x)$
3. Product: $(f \cdot g)(x) = f(x) \cdot g(x)$
4. Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad \text{provided } g(x) \neq 0$

Domain Rule: The domain of $(f + g)$, $(f - g)$, and $(f \cdot g)$ is the intersection of their individual domains: $\text{Domain}(f) \cap \text{Domain}(g)$. For the quotient $\frac{f}{g}$, we must additionally exclude all $x$ where $g(x) = 0$.

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4. Composite Functions

A composite function $(f \circ g)(x) = f(g(x))$ is formed by feeding the output of an inner function $g(x)$ directly as the input into an outer function $f(x)$.

Input x  --->  [ Inner Function g ]  --->  g(x)  --->  [ Outer Function f ]  --->  f(g(x))

Domain of Composite Functions

The domain of $(f \circ g)(x)$ consists of all $x$ in the domain of $g$ such that the output $g(x)$ lies within the domain of $f$.

Two-Step Algorithm to Find Domain of $(f \circ g)(x)$:

1. Find the domain of the inner function $g(x)$ and state its restrictions.
2. Construct the algebraic formula $f(g(x))$ and find any additional restrictions introduced by the outer structure.

Worked Example 2.2C: Composite Function and Domain Determination

Given $f(x) = \frac{4}{x - 3}$ and $g(x) = \sqrt{x + 2}$. Find $(f \circ g)(x)$ and state its domain.

Step 1: Determine domain of inner function $g(x)$. $x + 2 \ge 0 \implies x \ge -2$

Step 2: Form composite expression $(f \circ g)(x)$. $(f \circ g)(x) = f(g(x)) = f(\sqrt{x + 2}) = \frac{4}{\sqrt{x + 2} - 3}$

Step 3: Find restrictions on outer function. The denominator cannot equal zero: $\sqrt{x + 2} - 3 \neq 0 \implies \sqrt{x + 2} \neq 3 \implies x + 2 \neq 9 \implies x \neq 7$

Step 4: Combine inner domain $x \ge -2$ with restriction $x \neq 7$. $\text{Domain: } [-2, 7) \cup (7, \infty)$

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5. CLEP Precalculus Exam Traps