5.6 Function Notation, Composition & Transformations
Key Takeaways
- Interpret function notation f(x) as an input-output mapping rule: evaluate f at numerical constants and algebraic expressions by substituting the entire argument into every occurrence of x.
- Evaluate composite functions (f ∘ g)(x) = f(g(x)) from the inside out: evaluate inner function g(x) first, then feed that output directly into outer function f.
- Master rigid translations: f(x) + c shifts UP by c, f(x) - c shifts DOWN by c, f(x - c) shifts RIGHT by c, and f(x + c) shifts LEFT by c.
- Master non-rigid stretches, compressions, and reflections: a · f(x) scales vertically (and reflects across x-axis if a < 0); f(cx) scales horizontally by factor 1/c (and reflects across y-axis if c < 0).
- Track coordinate transformations using point mapping: under g(x) = a · f(c(x - h)) + k, any original point (x, y) on f transforms to (x/c + h, a · y + k).
Function Notation, Composition & Transformations
Function notation and transformations are fundamental to advanced mathematics on the Digital SAT. Questions assess evaluating functions with algebraic inputs, computing nested composite functions $(f \circ g)(x)$, and predicting how algebraic modifications alter graphs geometrically.
1. Function Notation Fluency & Algebraic Evaluation
Function notation $f(x)$ reads "$f$ of $x$" and represents the output value produced when input $x$ is processed by the function rule.
The "Empty Box" Substitution Method:
To avoid algebraic errors when evaluating $f(\text{expression})$, replace every instance of the variable with parenthesized empty boxes before inserting the input:
If $f(x) = 2x^2 - 5x + 3$:
- Numerical Input: $f(-3) = 2(-3)^2 - 5(-3) + 3 = 2(9) + 15 + 3 = 36$
- Algebraic Input: $f(2a - 1) = 2(2a - 1)^2 - 5(2a - 1) + 3$
+-----------------------------------------------------------------------------+
| FUNCTION INPUT SUBSTITUTION |
| |
| Given: f(x) = x^2 - 3x |
| |
| Evaluate f(x + 2): |
| Step 1: Replace x with parentheses: (x + 2)^2 - 3(x + 2) |
| Step 2: Expand binomial square: x^2 + 4x + 4 - 3x - 6 |
| Step 3: Combine like terms: x^2 + x - 2 |
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2. Composite Functions: $(f \circ g)(x) = f(g(x))$
A composite function applies one function to the output of another. The notation $(f \circ g)(x)$ is defined as $f(g(x))$.
+-----------------------------------------------------------------------------+
| COMPOSITE FUNCTION FLOWCHART |
| |
| Input (x) ====> [ INNER FUNCTION: g(x) ] |
| │ |
| v Output g(x) |
| [ OUTER FUNCTION: f( ) ] |
| │ |
| v |
| Final Output: f(g(x)) |
+-----------------------------------------------------------------------------+
Key Principles of Composition:
- Work Inside Out: Always evaluate the inner function $g(x)$ first. Then use that result as the input for the outer function $f$.
- Non-Commutative: In general, $f(g(x)) \ne g(f(x))$. Order of operations is paramount.
Example with Tabular Data:
| $x$ | $f(x)$ | $g(x)$ |
|---|---|---|
| $1$ | $4$ | $3$ |
| $2$ | $1$ | $5$ |
| $3$ | $2$ | $1$ |
| $5$ | $3$ | $2$ |
- To find $f(g(2))$: First find $g(2) = 5$. Then find $f(5) = 3$. Thus, $f(g(2)) = 3$.
- To find $g(f(2))$: First find $f(2) = 1$. Then find $g(1) = 3$. Thus, $g(f(2)) = 3$.
3. Master Transformation Rules
Transformations modify the position, size, or orientation of a parent graph $y = f(x)$.
| Transformation Form | Operation Type | Direction / Effect | Coordinate Mapping $(x, y) \to$ |
|---|---|---|---|
| $y = f(x) + c$ | Vertical Shift | Shift UP by $c$ units | $(x, y + c)$ |
| $y = f(x) - c$ | Vertical Shift | Shift DOWN by $c$ units | $(x, y - c)$ |
| $y = f(x - c)$ | Horizontal Shift | Shift RIGHT by $c$ units | $(x + c, y)$ |
| $y = f(x + c)$ | Horizontal Shift | Shift LEFT by $c$ units | $(x - c, y)$ |
| $y = a \cdot f(x) ; (a > 1)$ | Vertical Stretch | Stretch vertically by factor $a$ | $(x, a \cdot y)$ |
| $y = a \cdot f(x) ; (0 < a < 1)$ | Vertical Compress | Compress vertically by factor $a$ | $(x, a \cdot y)$ |
| $y = -f(x)$ | Reflection | Reflect across $x$-axis (flips upside down) | $(x, -y)$ |
| $y = f(-x)$ | Reflection | Reflect across $y$-axis (flips left/right) | $(-x, y)$ |
| $y = f(c x) ; (c > 1)$ | Horizontal Compress | Compress horizontally by factor $1/c$ | $(x/c, y)$ |
| $y = f(c x) ; (0 < c < 1)$ | Horizontal Stretch | Stretch horizontally by factor $1/c$ | $(x/c, y)$ |
+-----------------------------------------------------------------------------+
| HORIZONTAL VS. VERTICAL SHIFT RULES |
| |
| OUTSIDE MODIFICATION = VERTICAL (Direct): |
| f(x) + 4 ====> UP 4 units f(x) - 4 ====> DOWN 4 units |
| |
| INSIDE MODIFICATION = HORIZONTAL (Counter-Intuitive / Inverted): |
| f(x - 4) ====> RIGHT 4 units f(x + 4) ====> LEFT 4 units |
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[!WARNING] The Horizontal Shift Sign Trap: Inside the function parentheses, operations behave in the opposite direction of their sign:
- $f(x - 5)$ moves the graph RIGHT $+5$ units.
- $f(x + 5)$ moves the graph LEFT $-5$ units.
- To find where the new feature lands, set the interior expression equal to the original coordinate: $x - 5 = 0 \implies x = 5$.
4. Multi-Step Transformations & Point Mapping
When combining multiple transformations in the general form $g(x) = a \cdot f(c(x - h)) + k$, use the Point Mapping Formula:
Example Transformation Walkthrough:
Suppose $y = f(x)$ has vertex $(3, -4)$. What are the coordinates of the corresponding point on $g(x) = -2 f(x + 4) + 7$?
- Horizontal transformation: $x + 4 = 3 \implies x = 3 - 4 = -1$.
- Vertical transformation: $y_{\text{new}} = -2(-4) + 7 = 8 + 7 = 15$.
- New point on $g(x)$ is $(-1, 15)$.
5. Step-by-Step Worked SAT Exam Examples
Worked Example 1: Nested Composite Algebraic Evaluation
Problem: Given $f(x) = 3x - 5$ and $g(x) = x^2 + 2x$, what is the value of $g(f(3)) - f(g(2))$?
Step-by-Step Solution:
- Evaluate inner term $f(3)$ for the first expression:
- Evaluate outer term $g(4)$:
- Evaluate inner term $g(2)$ for the second expression:
- Evaluate outer term $f(8)$:
- Compute the difference:
Worked Example 2: Determining Vertex Shift Under Transformation
Problem: The quadratic function $f(x) = (x - 6)^2 + 8$ has a vertex at $(6, 8)$. If function $h$ is defined by $h(x) = f(x + 3) - 5$, what is the minimum value of $h(x)$ and at what value of $x$ does it occur?
Step-by-Step Solution:
- Analyze the horizontal shift: $f(x + 3)$ shifts the graph $3$ units to the left: $x_{\text{vertex}} = 6 - 3 = 3$.
- Analyze the vertical shift: $- 5$ shifts the graph $5$ units down: $y_{\text{vertex}} = 8 - 5 = 3$.
- State the minimum: Since the parabola opens upward ($a = 1 > 0$), the minimum value of $h(x)$ is $3$, occurring at $x = 3$.
6. Desmos Testing Strategies for Functions & Transformations
[!TIP] Defining Functions and Nested Compositions in Desmos: Desmos supports direct function definition and composite evaluation:
- Type
f(x) = 2x^2 - 3x + 1on line 1.- Type
g(x) = 3x - 2on line 2.- Type
f(g(2)) - g(f(2))on line 3. Desmos will calculate the final numeric answer directly without manual substitution.
[!NOTE] Visualizing Graph Transformations: To check a transformation like $g(x) = -2f(x + 4) + 7$:
- Plot the parent function
f(x) = (x - 3)^2 - 4.- On line 2, plot
g(x) = -2 * f(x + 4) + 7.- Click the vertex of both curves to confirm that $(3, -4)$ transformed into $(-1, 15)$.
Given $f(x) = 2x^2 - 3x + 1$ and $g(x) = 3x - 2$, what is the value of $f(g(2)) - g(f(2))$?
The graph of $y = f(x)$ contains the point $(-3, 8)$. If the function $g$ is defined by $g(x) = 4f(x + 5) - 6$, which point must lie on the graph of $y = g(x)$?
The function $f(x)$ has a minimum value of $-7$ at $x = 4$. If the function $h(x)$ is defined by $h(x) = -3 f(x - 2) + 5$, what is the maximum value of $h(x)$ and at what $x$-value does it occur?