Function Notation and Transformations

Function Notation Review

Basic Notation

$f(x) = 2x + 3$ means "the function $f$ takes an input $x$ and outputs $2x + 3$"

Examples:

• $f(5) = 2(5) + 3 = 13$
• $f(-2) = 2(-2) + 3 = -1$
• $f(a) = 2a + 3$

Composite Functions

Notation: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = x^2$ and $g(x) = x + 1$:

$f(g(3)) = f(3 + 1) = f(4) = 4^2 = 16$

$g(f(3)) = g(3^2) = g(9) = 9 + 1 = 10$

Note: Order matters! $f(g(x)) \neq g(f(x))$

Inverse Functions

Notation: $f^{-1}(x)$

Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$

Verify: $f(f^{-1}(7)) = f(\frac{7-3}{2}) = f(2) = 2(2) + 3 = 7$ ✓

Function Transformations

Vertical Transformations

Vertical Shift:

• $f(x) + k$ shifts UP $k$ units
• $f(x) - k$ shifts DOWN $k$ units

Vertical Stretch/Compression:

• $a \cdot f(x)$ where $a > 1$: stretch (taller)
• $a \cdot f(x)$ where $0 < a < 1$: compression (shorter)

Reflection over x-axis:

• $-f(x)$ flips the graph upside down

Horizontal Transformations

Horizontal Shift:

• $f(x - h)$ shifts RIGHT $h$ units (opposite of what you'd think!)
• $f(x + h)$ shifts LEFT $h$ units

Horizontal Stretch/Compression:

• $f(bx)$ where $b > 1$: compression (narrower)
• $f(bx)$ where $0 < b < 1$: stretch (wider)

Reflection over y-axis:

• $f(-x)$ flips the graph horizontally

Transformation Examples

Example 1: Multiple Transformations

Given: $f(x) = x^2$
New: $g(x) = -2f(x - 3) + 1 = -2(x-3)^2 + 1$

Transform in this order:

1. Shift right 3: $(x-3)^2$
2. Stretch vertically by 2: $2(x-3)^2$
3. Reflect over x-axis: $-2(x-3)^2$
4. Shift up 1: $-2(x-3)^2 + 1$

Example 2: From Graph to Equation

If the parent function $f(x) = \sqrt{x}$ is:

• Shifted left 2
• Reflected over x-axis
• Shifted down 3

Equation: $g(x) = -\sqrt{x + 2} - 3$

SAT Question Types

Type 1: Evaluate Composite Functions

Given: $f(x) = x^2 + 1$ and $g(x) = 2x - 3$

Find: $f(g(2))$

Solution:

1. Find $g(2) = 2(2) - 3 = 1$
2. Find $f(1) = 1^2 + 1 = 2$

Type 2: Match Transformations to Graphs

Strategy:

• Check key points (vertex, intercepts)
• Identify shifts first (easiest to spot)
• Then check reflections and stretches

Type 3: Inverse Function Properties

If $f(5) = 12$, what is $f^{-1}(12)$?

Answer: $f^{-1}(12) = 5$ (inverse "undoes" the function)

Common SAT Mistakes

❌ Confusing $f(x) + 3$ (shift up) with $f(x + 3)$ (shift left)
❌ Thinking $f(x - 2)$ shifts left (it shifts RIGHT!)
❌ Forgetting order matters in composite functions
❌ Not simplifying composite functions step-by-step

Transformation Quick Reference

| Transformation | Notation | Effect | |---------------|----------|--------| | Shift up $k$ | $f(x) + k$ | Move graph up | | Shift down $k$ | $f(x) - k$ | Move graph down | | Shift right $h$ | $f(x - h)$ | Move graph right | | Shift left $h$ | $f(x + h)$ | Move graph left | | Reflect over x-axis | $-f(x)$ | Flip upside down | | Reflect over y-axis | $f(-x)$ | Flip horizontally | | Vertical stretch | $a \cdot f(x)$, $a>1$ | Make taller | | Vertical compression | $a \cdot f(x)$, $0<a<1$ | Make shorter |

Pro Tips

✓ Inside the parentheses affects x (horizontal)
✓ Outside the parentheses affects y (vertical)
✓ Horizontal shifts are counterintuitive (opposite sign)
✓ Work from inside out for composite functions