5.6 Function Notation, Composition & Transformations

Q: Given f(x) = 2x^2 - 3x + 1 and g(x) = 3x - 2, what is the value of f(g(2)) - g(f(2))?

14. Evaluate f(g(2)) first: calculate inner function g(2) = 3(2) - 2 = 4. Then evaluate f(4) = 2(4)^2 - 3(4) + 1 = 2(16) - 12 + 1 = 32 - 12 + 1 = 21. Next, evaluate g(f(2)): calculate inner function f(2) = 2(2)^2 - 3(2) + 1 = 2(4) - 6 + 1 = 8 - 6 + 1 = 3. Then evaluate g(3) = 3(3) - 2 = 9 - 2 = 7. Finally, compute the difference: f(g(2)) - g(f(2)) = 21 - 7 = 14.

Q: 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)?

(-8, 26). To evaluate g(x) = 4f(x + 5) - 6 using the known point f(-3) = 8, the interior argument of f must equal -3. Set x + 5 = -3, which solves to x = -8. Substituting x = -8 into g(x) yields g(-8) = 4f(-8 + 5) - 6 = 4f(-3) - 6. Since f(-3) = 8, calculate g(-8) = 4(8) - 6 = 32 - 6 = 26. Therefore, the corresponding transformed point on g(x) is (-8, 26).

Q: 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?

Maximum value of 26 at x = 6. The original point is (4, -7), where f(4) = -7 is a minimum. For h(x) = -3f(x - 2) + 5, the input to f must be 4, so x - 2 = 4 \implies x = 6. The vertical transformation multiplies the output by -3 and adds 5. Because multiplying by a negative constant -3 reflects the graph across the x-axis, the minimum of f transforms into a maximum of h. Compute the transformed value: h(6) = -3 f(4) + 5 = -3(-7) + 5 = 21 + 5 = 26. Thus, h(x) reaches a maximum value of 26 at x = 6.

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).

Last updated: August 2026

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 $f(x) = 2x^2 - 5x + 3$ :

f(\Box) = 2(\Box)^2 - 5(\Box) + 3

Numerical Input: $f(-3) = 2(-3)^2 - 5(-3) + 3 = 2(9) + 15 + 3 = 36$
Algebraic Input: $f(2a - 1) = 2(2a - 1)^2 - 5(2a - 1) + 3$ $= 2(4a^2 - 4a + 1) - 10a + 5 + 3 = 8a^2 - 8a + 2 - 10a + 8 = 8a^2 - 18a + 10$

+-----------------------------------------------------------------------------+
|                        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                         |
+-----------------------------------------------------------------------------+

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       |
+-----------------------------------------------------------------------------+

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:

\text{Original Point on } f(x): (x, y) \implies \text{Transformed Point on } g(x): \left(\frac{x}{c} + h, \; a \cdot y + k\right)

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: $f(3) = 3(3) - 5 = 9 - 5 = 4$
Evaluate outer term $g(4)$ : $g(4) = (4)^2 + 2(4) = 16 + 8 = 24 \implies g(f(3)) = 24$
Evaluate inner term $g(2)$ for the second expression: $g(2) = (2)^2 + 2(2) = 4 + 4 = 8$
Evaluate outer term $f(8)$ : $f(8) = 3(8) - 5 = 24 - 5 = 19 \implies f(g(2)) = 19$
Compute the difference: $g(f(3)) - f(g(2)) = 24 - 19 = 5$

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 + 1 on line 1.
Type g(x) = 3x - 2 on 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 $(-1, 15)$ .

Test Your Knowledge

Given $f(x) = 2x^2 - 3x + 1$ and $g(x) = 3x - 2$ , what is the value of $f(g(2)) - g(f(2))$ ?

$-8$

$10$

$14$

$28$

Test Your Knowledge

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 $y = g(x)$ ?

$(-8, 26)$

$(2, 26)$

$(-8, 38)$

$(2, 2)$

Test Your Knowledge

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?

Minimum value of $-16$ at $x = 2$

Maximum value of $26$ at $x = 2$

Minimum value of $-26$ at $x = 6$

Maximum value of $26$ at $x = 6$

Sections you finish are checked off in the contents.

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6.1 Ratios, Proportions & Dimensional Analysis

Chapter 6: Problem-Solving and Data Analysis

SAT Math

5.6 Function Notation, Composition & Transformations

Key Takeaways

Function Notation, Composition & Transformations

1. Function Notation Fluency & Algebraic Evaluation

The "Empty Box" Substitution Method:

2. Composite Functions: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f∘g)(x)=f(g(x))

Key Principles of Composition:

Example with Tabular Data:

3. Master Transformation Rules

4. Multi-Step Transformations & Point Mapping

Example Transformation Walkthrough:

5. Step-by-Step Worked SAT Exam Examples

Worked Example 1: Nested Composite Algebraic Evaluation

Worked Example 2: Determining Vertex Shift Under Transformation

6. Desmos Testing Strategies for Functions & Transformations

2. Composite Functions: $(f \circ g)(x) = f(g(x))$