🎯⭐ INTERACTIVE LESSON

Functions

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Functions - Complete Interactive Lesson

Part 1: Function Notation

Functions & Graphs

Part 1 of 7 — Function Notation & Evaluation

What is a Function?

A function $f$ assigns exactly one output to each input. Written as $f(x) = \text{expression}$ .

$f(3)$ means "plug in $x = 3$ "
$f(a + 1)$ means "replace every $x$ with "

Example: If $f(x) = 2x^2 - 3x + 1$ :

$f(4) = 2(16) - 12 + 1 = 21$

$f(-1) = 2(1) - 3(-1) + 1 = 2 + 3 + 1 = 6$

Domain & Range

Domain: all valid input values (check for division by zero, square roots of negatives)
Range: all possible output values

SAT Function Notation Tricks

$f(x) = 3$ asks: "For what value(s) of $x$ does the output equal 3?" This means solving $f(x) = 3$ , NOT evaluating $f(3)$ .

On a graph: find where $y = 3$ intersects the curve.

Function Evaluation 🎯

Key Takeaways — Part 1

$f(a)$ = substitute $a$ for every $x$ in the function
" $f(x) = k$ " means solve for , not evaluate at

Part 2: Interpreting Graphs

Functions & Graphs

Part 2 of 7 — Composite and Inverse Functions

Composition: $f(g(x))$

"Evaluate inside out" — first compute $g(x)$ , then plug the result into $f$ .

Example: $f(x) = x^2$ and

Part 3: Domain & Range

Functions & Graphs

Part 3 of 7 — Transformations of Functions

Vertical Transformations (Outside the function)

Transformation	Equation	Effect
Shift up $k$	$f(x) + k$	Graph moves up $k$ units
Shift down

Part 4: Transformations

Functions & Graphs

Part 4 of 7 — Piecewise & Absolute Value Functions

Piecewise Functions

A function defined by different rules for different parts of its domain:

$f(x) = \begin{cases} x + 3 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

Part 5: Function Composition

Functions & Graphs

Part 5 of 7 — Graph Analysis & Interpretation

Increasing vs. Decreasing

Increasing: as $x$ moves right, $y$ goes up
Decreasing: as $x$ moves right, $y$ goes down
Constant: horizontal line segment

Maximum and Minimum Values

Absolute max/min: the highest/lowest y-value on the entire graph
Relative (local) max/min: higher/lower than nearby points

On the SAT, these appear as:

Part 6: Problem-Solving Workshop

Functions & Graphs

Part 6 of 7 — Even/Odd Functions and Symmetry

Even Functions: $f(-x) = f(x)$

Symmetric about the y-axis
Examples: $x^2$ , $|x|$ ,

Part 7: Review & Applications

Functions & Graphs

Part 7 of 7 — Review & SAT-Level Mixed Practice

Functions Cheat Sheet

Concept	Key Idea
$f(a)$	Substitute $a$ into the function
$f(x) = k$

Domain restrictions: denominators ≠ 0, expressions under even roots ≥ 0

On graphs:

f(a) = b

means the point

(a, b)

is on the curve

$f(g(2)) = f(5) = 25$

$g(f(2)) = g(4) = 7$ — order matters!

Inverse Functions: $f^{-1}(x)$

If $f(a) = b$ , then $f^{-1}(b) = a$ . Inverses "undo" each function.

To find $f^{-1}(x)$ :

Replace $f(x)$ with $y$
Swap $x$ and $y$
Solve for $y$

Example: $f(x) = 2x + 3$

$y = 2x + 3$
Swap: $x = 2y + 3$
Solve: $y = (x - 3)/2$
$f^{-1}(x) = \frac{x - 3}{2}$

The graph of $f^{-1}$ is the reflection of $f$ across the line $y = x$ .

Composition & Inverses 🎯

Key Takeaways — Part 2

Composition: evaluate inside out — $f(g(x))$ ≠ $g(f(x))$ in general
Inverse: swap $x$ and $y$ , then solve for $y$
If $f(a) = b$ , then $f^{-1}(b) = a$
The graph of $f^{-1}$ is the reflection of $f$ over $y = x$

Horizontal Transformations (Inside the function)

Transformation	Equation	Effect
Shift right $h$	$f(x - h)$	Graph moves right
Shift left $h$	$f(x + h)$	Graph moves left
Stretch by $1/b$	$f(bx)$	Graph gets wider
Compress by $b$	$f(bx)$	Graph gets narrower
Reflect over y-axis	$f(-x)$	Flip left-right

Horizontal transformations are opposite to what you might expect:

$f(x - 3)$ moves the graph right, not left
$f(2x)$ makes the graph narrower, not wider

Transformations 🎯

Key Takeaways — Part 3

Outside the function ( $+k$ , $\times a$ , $-$ ): vertical changes, work as expected
Inside the function ( $x - h$ , $bx$ , $-x$ ): horizontal changes, opposite direction
$f(x - h) + k$ : shifts right $h$ and up $k$
Vertical stretches don't move the vertex; they affect the shape

$f(-2) = -2 + 3 = 1$ (use first rule since $-2 < 0$ )

$f(3) = 9$ (use second rule since $3 \geq 0$ )

Absolute Value as Piecewise

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Graphing $y = a|x - h| + k$

V-shaped graph with vertex at $(h, k)$
Opens up if $a > 0$ , opens down if $a < 0$
Slope of right branch is $a$ , left branch is $-a$

When the SAT shows a piecewise graph, read each segment separately. Check:

What's the y-value at specific x-values?
Are the endpoints open circles (excluded) or closed circles (included)?

Piecewise Functions 🎯

Key Takeaways — Part 4

Piecewise functions: check which condition $x$ satisfies, then use that rule
Absolute value makes a V-shape with vertex at $(h, k)$
Open circle = point not included; closed circle = point included
Check boundary values carefully — which piece applies at the boundary?

"Over which interval is $f$ increasing?"
"At what value of $x$ does $f$ attain its maximum?"
"What is the maximum value of $f$ ?" (asking for the y-coordinate)

Average rate of change from $x = a$ to $x = b$ :

$\text{Rate} = \frac{f(b) - f(a)}{b - a}$

This is just the slope of the secant line between two points.

x-intercepts: where $f(x) = 0$ (solve or read from graph)
y-intercept: evaluate $f(0)$ (or read where graph crosses y-axis)

Graph Analysis 🎯

Key Takeaways — Part 5

Increasing/decreasing: determined by the direction of the y-values as x increases
Average rate of change $= \frac{f(b) - f(a)}{b - a}$ = slope of secant line
x-intercepts: $f(x) = 0$ ; y-intercept: $f(0)$
"Maximum value of $f$ " = the y-coordinate, not the x-coordinate

(3, 5)

is on the graph, then

(-3, 5)

is too

Odd Functions: $f(-x) = -f(x)$

Symmetric about the origin (180° rotation)
Examples: $x^3$ , $x$ , $\sin(x)$
If $(3, 5)$ is on the graph, then $(-3, -5)$ is too

Testing Algebraically

For $f(x) = x^4 - 3x^2$ : $f(-x) = (-x)^4 - 3(-x)^2 = x^4 - 3x^2 = f(x)$ → even

For $g(x) = x^3 + x$ : $g(-x) = -x^3 - x = -(x^3 + x) = -g(x)$ → odd

Neither Even nor Odd

$h(x) = x^2 + x$ : $h(-x) = x^2 - x \neq h(x)$ and $\neq -h(x)$ → neither

Symmetry questions often appear as: "If the graph of $f$ is symmetric about the y-axis and $f(3) = 7$ , what is $f(-3)$ ?" Answer: $f(-3) = 7$ .

Key Takeaways — Part 6

Even: $f(-x) = f(x)$ → symmetric about y-axis
Odd: $f(-x) = -f(x)$ → symmetric about origin
Not all functions are even or odd — most are neither
To test: substitute $-x$ and compare to $f(x)$ and $-f(x)$

SAT Strategies for Function Questions

Use the answer choices — if asked for a function and given formulas, test with a value
Read graphs carefully — pay attention to open vs. closed circles
Don't confuse $f(a) = b$ with $f(b) = a$ — this is the inverse trap
For word problems: identify what variable is the input and what is the output

Mixed Review 🎯

Key Takeaways — Part 7

Master function notation: $f(x)$ is the output for input $x$
Composition: always evaluate innermost function first
Transformations: inside = horizontal (opposite direction), outside = vertical (same direction)
Domain restrictions come from denominators = 0 and even roots of negatives
On the SAT, graph reading and function notation are equally important

Functions

Inverse Functions: $f^{-1}(x)$

Graph of Inverse

Key Takeaways — Part 2

Horizontal Transformations (Inside the function)

Key Insight ⚠️

Key Takeaways — Part 3

Absolute Value as Piecewise

Graphing $y = a|x - h| + k$

SAT Graph Reading

Key Takeaways — Part 4

Rate of Change

Intercepts

Key Takeaways — Part 5

Odd Functions: $f(-x) = -f(x)$

Testing Algebraically

Neither Even nor Odd

SAT Connection

Key Takeaways — Part 6

SAT Strategies for Function Questions

Key Takeaways — Part 7

Functions

Functions - Complete Interactive Lesson

Part 1: Function Notation

Functions & Graphs

What is a Function?

Domain & Range

SAT Function Notation Tricks

Key Takeaways — Part 1

Part 2: Interpreting Graphs

Functions & Graphs

Composition: f(g(x))f(g(x))f(g(x))

Part 3: Domain & Range

Functions & Graphs

Vertical Transformations (Outside the function)

Part 4: Transformations

Functions & Graphs

Piecewise Functions

Part 5: Function Composition

Functions & Graphs

Increasing vs. Decreasing

Maximum and Minimum Values

Part 6: Problem-Solving Workshop

Functions & Graphs

Even Functions: f(−x)=f(x)f(-x) = f(x)f(−x)=f(x)

Part 7: Review & Applications

Functions & Graphs

Functions Cheat Sheet

Inverse Functions: f−1(x)f^{-1}(x)f−1(x)

Graph of Inverse

Key Takeaways — Part 2

Horizontal Transformations (Inside the function)

Key Insight ⚠️

Key Takeaways — Part 3

Absolute Value as Piecewise

Graphing y=a∣x−h∣+ky = a|x - h| + ky=a∣x−h∣+k

SAT Graph Reading

Key Takeaways — Part 4

Rate of Change

Intercepts

Key Takeaways — Part 5

Odd Functions: f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x)

Testing Algebraically

Neither Even nor Odd

SAT Connection

Key Takeaways — Part 6

SAT Strategies for Function Questions

Key Takeaways — Part 7

Composition: $f(g(x))$

Even Functions: $f(-x) = f(x)$

Inverse Functions: $f^{-1}(x)$

Graphing $y = a|x - h| + k$

Odd Functions: $f(-x) = -f(x)$