Functions & Symmetry Master Class

The Problem with "y"

Imagine you are controlling a rocket. You need to track two things:

Height: $y = 10t^2$
Speed: $y = 20t$

If someone asks "What is $y$ when $t=3$?", you would be confused. Which $y$? Height or Speed?

The Solution: Name Tags

We give each rule a unique name. This avoids confusion.

$h(t) = 10t^2$ (h for Height)
$v(t) = 20t$ (v for Velocity)

Now, if I ask for $h(3)$, you know exactly which rule to use.

🚀 Rocket Dashboard

Time ($t$ seconds):

Height $h(t)$

90

$10(3)^2$

Velocity $v(t)$

60

$20(3)$

Notice how $h(3)$ and $v(3)$ give different answers for the same input. "y" couldn't handle this!

What is a Function?

Don't overcomplicate it. A function is just a Processor.

It takes an Input, follows a Rule, and gives an Output.

🍎

Input (x)

→

⚙️

f( )

→

🧃

Output f(x)

If the rule is "Juice it", then $f(\text{Apple}) = \text{Apple Juice}$.

The "Empty Box" Machine

In algebra, $f(x)$ is a machine with an empty slot: $f(\Box)$.

Whatever you drop in the slot replaces every instance of $x$ in the rule.

The Rule:
$f(x) = 2x^2 - 5$

Waiting for input...

Challenge: Complex Inputs

If the rule is $f(x) = x^2 + 1$, what is $f(1-x)$?

(We replace the "Apple" with a "Basket of Fruit". The whole basket gets processed.)

Symmetry in Algebra

Even Function: Mirror Symmetry across y-axis.
Rule: $f(-x) = f(x)$. (Example: $x^2$)
Odd Function: Rotational Symmetry (180°) around origin.
Rule: $f(-x) = -f(x)$. (Example: $x^3$)

Algebraic Testing Lab

Test 1 (Numeric): If $f(x)$ is Even and $f(2) = 4$, what is $f(-2)$?

Test 2 (Algebraic): Classify the expression $x^2 + 5$. (Test by putting $-x$ in)

The Shifted Symmetry Line

Standard Even symmetry is around the line $x=0$. We check distance $x$ from 0:
$f(0-x) = f(0+x) \rightarrow f(-x) = f(x)$

What if the mirror moves to $x=1$?

We check distance $x$ from the new center 1:

$$ f(1-x) = f(1+x) $$

This means the graph is symmetrical around the vertical line $x=1$.

Quick Quiz: What does $f(2-x) = f(2+x)$ mean?

Point Symmetry (Odd Shift)

Standard Odd functions rotate around the origin point $(0,0)$.

Rule: $f(-x) = -f(x)$.

If we shift the center of rotation to $x=a$ (a point $(a,0)$), the logic holds:

$$ f(a-x) = -f(a+x) $$

This describes a graph that rotates 180° around the point $(a,0)$.

Logic Puzzles (No Series)

These questions test the same logic as the Scholarship question (using symmetry to find values) but without long summations.

Puzzle 1: The Odd Mirror

Let $f(x)$ be an Odd function.

This means $f(-x) = -f(x)$.

If $f(-3) = 4$, what is the value of $f(3)$?

Puzzle 2: The Shifted Symmetry

A function satisfies the rule: $f(2 - x) = f(2 + x)$.

(This means the graph is symmetrical around the line $x=2$)

If we know $f(0) = 7$, what is the value of $f(4)$?

Hint: Test $x=2$ in the formula. $f(2-2) = f(2+2)$.

Puzzle 3: The Combined Challenge

Let $f(x)$ be an Odd function. It also satisfies the symmetry rule:

$f(1-x) = f(1+x)$

Given that $f(1) = 5$. Find the value of $f(3)$.

Steps to solve:

$f(x)$ is Odd, so $f(-x) = -f(x)$.
Use the symmetry rule with $x=2$: $f(1-2) = f(1+2)$.
Simplify that expression to relate $f(-1)$ and $f(3)$.
Use the Odd property to find $f(-1)$ from $f(1)$.

f(3) =

Visual Confirmation

This graph shows an example function $f(x) = 5\sin(\frac{\pi x}{2})$ that satisfies the puzzle's conditions.

It is Odd, passes through $(1,5)$, and has symmetry around $x=1$. Check the point at $x=3$!