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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.

Worked Example 1

If $f(x) = x^2 + 3x - 5$ , evaluate $f(a + 2)$ .

Step	Work
Replace $x$ with $(a+2)$	$f(a+2) = (a+2)^2 + 3(a+2) - 5$

Worked Example 2

The domain of $f(x) = \sqrt{2x - 6}$ .

Step	Work
Expression under radical $\geq 0$	$2x - 6 \geq 0$
Solve	$x \geq 3$

Function Evaluation 🎯

Function Evaluation with Tables

The SAT often gives a table and asks you to evaluate:

$x$	$f(x)$
$-1$	$4$
$0$

Harder Function Notation 🎯

What Does the Notation Mean? 🔍

For each expression, choose what it represents.

Key Takeaways — Part 1

Notation	Meaning
$f(a)$	Substitute $a$ for every $x$
$f(x) = k$

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
$f(x) = k$

From this table: $f(1) = 0$ , $f(0) = 2$ , and $f(f(1)) = f(0) = 2$ .

Worked Example 3 — "Solving" $f(x) = k$

Using the table above, for what values of $x$ is $f(x) = 0$ ?

Step	Work
Scan the $f(x)$ column for $0$	$f(1) = 0$ and $f(3) = 0$
Answer	$x = 1$ and $x = 3$

SAT Trap: If the question asks " $f(x) = 0$ ", do NOT evaluate $f(0) = 2$ . Find where the output is $0$ .

On graphs: $f(a) = b$ means the point $(a, b)$ is on the curve
For expressions like $f(a+1)$ , replace every $x$ with $(a+1)$ , then simplify

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

$g(f(2)) = g(4) = 7$ — 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 $y$

Find $f^{-1}(x)$ for $f(x) = \frac{2x + 1}{3}$ .

Step	Work
Set $y = \frac{2x + 1}{3}$
Swap $x$ and $y$	$x = \frac{2y + 1}{3}$
Multiply by 3	$3x = 2y + 1$
Solve for $y$	$y = \frac{3x - 1}{2}$
Result	$f^{-1}(x) = \frac{3x - 1}{2}$

If $f(x) = x^2 + 1$ and $g(x) = 3x - 2$ , find $f(g(x))$ .

Step	Work
Start with $g(x)$	$g(x) = 3x - 2$
Plug into $f$	$f(3x - 2) = (3x-2)^2 + 1$
Expand	$= 9x^2 - 12x + 4 + 1$
Simplify	$= 9x^2 - 12x + 5$

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

Composition & Inverses 🎯

Composition with Tables

The SAT frequently gives two tables and asks for a composition:

$x$	$f(x)$
1	3
2	5
3	1

$x$	$g(x)$
1	2
2	3
3	1

Find $f(g(2))$ : $g(2) = 3$ , then $f(3) = 1$ . Answer: .

Find $g(f(1))$ : $f(1) = 3$ , then $g(3) = 1$ . Answer: .

Verifying Inverses

Two functions are inverses if $f(g(x)) = x$ AND $g(f(x)) = x$ .

Example: $f(x) = 2x + 3$ and $g(x) = \frac{x - 3}{2}$

$f(g(x)) = 2\left(\frac{x-3}{2}\right) + 3 = (x - 3) + 3 = x$ ✓

Harder Composition 🎯

Composition Order Matters! 🔍

Given $f(x) = x + 1$ and $g(x) = 2x$ , evaluate each.

Key Takeaways — Part 2

Concept	Key Rule
$f(g(x))$	Evaluate inside out — order matters!
$f^{-1}(x)$	Swap $x$ and $y$ , solve for $y$
$f(a) = b$ → $f^{-1}(b) = a$	Inverse swaps input/output
Verify inverses	$f(g(x)) = x$ AND $g(f(x)) = x$
Graph of $f^{-1}$	Reflection of $f$ over $y = x$
Tables	Look up values step by step

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
Compress by $b$	$f(bx)$	Graph gets narrower ( $b > 1$ )
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

The graph of $y = f(x)$ passes through $(2, 5)$ . Where does the point move under $y = 3f(x - 4) + 1$ ?

Transformation	Effect on $(2, 5)$
$f(x - 4)$ : right 4	$(2 + 4, 5) = (6, 5)$
$3f$ : stretch $y$ by 3	$(6, 15)$
$+ 1$ : up 1	$(6, 16)$

The point moves to $(6, 16)$ .

Transformations 🎯

Combining Multiple Transformations

Apply transformations in this order:

Horizontal shifts and stretches (inside)
Reflections
Vertical stretches (outside)
Vertical shifts (outside)

Worked Example 2

Describe the transformations from $y = |x|$ to $y = -2|x + 3| + 7$ .

Piece	Transformation
$x + 3$	Shift left 3
$-2$ (coefficient)	Reflect over x-axis, stretch by factor 2
$+7$	Shift up 7
Vertex	Moves from $(0, 0)$ to

Worked Example 3

If $f(3) = 10$ , what point must be on $y = f(x + 5) - 2$ ?

Step	Work
Original point	$(3, 10)$
$f(x + 5)$ : left 5	$x$ -coordinate: $3 - 5$

Applied Transformations 🎯

Name That Transformation 🔍

Identify the transformation applied to $y = f(x)$ .

Key Takeaways — Part 3

Modification	Location	Direction
$f(x) + k$	Outside	Up $k$ (as expected)
$f(x - h)$	Inside	Right $h$ (opposite!)
$af(x)$	Outside	Vertical stretch/compress
$f(bx)$	Inside	Horizontal compress (opposite!)
$-f(x)$	Outside	Reflect over x-axis
$f(-x)$	Inside	Reflect over y-axis

To track a point: apply horizontal changes to $x$ , then vertical changes to $y$
Vertex transformations: $(0,0) → (h, k)$ in $a f(x - h) + k$

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

Evaluate $f(x) = \begin{cases} 2x - 1 & x \leq 3 \\ x^2 - 4 & x > 3 \end{cases}$ at $x = 3$ and $x = 5$ .

Input	Which rule?	Calculation	Result
$x = 3$	$3 \leq 3$ → first rule	$2(3) - 1$	$5$
$x = 5$	$5 > 3$ → second rule	$5^2 - 4$

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 🎯

Solving Absolute Value Equations

$|ax + b| = c$

Split into two cases: $ax + b = c$ or $ax + b = -c$ (only when $c \geq 0$ ).

Worked Example 2

Solve $|2x - 5| = 7$ .

Case	Equation	Solution
Positive	$2x - 5 = 7$	$x = 6$
Negative	$2x - 5 = -7$

Both solutions: $x = 6$ and $x = -1$ .

Worked Example 3

For what values of $x$ is $|x - 4| \leq 3$ ?

Step	Work
Remove absolute value	$-3 \leq x - 4 \leq 3$
Add 4 to all parts	$1 \leq x \leq 7$

SAT Tip: $|x - a| \leq d$ means " $x$ is within $d$ units of $a$ ." So $∣ x - 4$ means is within 3 of 4.

Absolute Value Equations 🎯

Piecewise or Absolute Value? 🔍

Classify each function type and identify key features.

Key Takeaways — Part 4

Concept	Key Rule
Piecewise	Check which condition $x$ satisfies, use that rule
$y = a	x - h
$	A
$	A
$	A
$	x - a
Open vs closed circles	Open = excluded, closed = included

"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.

The table shows values of $f$ . Find the average rate of change from $x = 1$ to $x = 5$ .

$x$	$1$	$2$	$3$	$4$	$5$
$f(x)$	$3$	$7$	$9$	$8$	$11$

Step	Work
Formula	$\frac{f(5) - f(1)}{5 - 1}$
Substitute	$\frac{11 - 3}{4}$
Simplify	$= 2$

Note: The function goes up and down between $x = 1$ and $x = 5$ , but the average rate of change only looks at endpoints.

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 🎯

Comparing Rates of Change

The SAT may ask you to compare rates of change over different intervals.

Worked Example 2

Using the table:

$x$	$0$	$1$	$2$	$3$	$4$
$f(x)$	$1$	$4$	$9$	$16$	$25$

Where is $f$ increasing fastest?

Interval	Rate of Change
$[0, 1]$	$(4 - 1)/1 = 3$
$[1, 2]$

$f$ increases fastest on $[3, 4]$ with rate $= 9$ .

(This pattern makes sense — it's $f(x) = (x+1)^2$ , a parabola that curves upward faster and faster.)

Positive, Negative, and Zero

Feature	Meaning on Graph
$f(x) > 0$	Graph is above the x-axis
$f(x) < 0$	Graph is below the x-axis
$f (x)$

Deeper Graph Analysis 🎯

Reading the Graph 🔍

For a function with $f(0) = 2$ , $f(2) = 6$ , $f(4) = 6$ , $f(6) = 0$ :

Key Takeaways — Part 5

Concept	Formula / Rule
Average rate of change	$\frac{f(b) - f(a)}{b - a}$ = slope of secant line
Increasing	$f$ goes up as $x$ increases
Decreasing	$f$ goes down as $x$ increases
$f(x) > 0$	Graph above x-axis
$f(x) = 0$	Graph on x-axis (x-intercept)
"Maximum value of $f$ "	The y-coordinate, not the x-coordinate

Zero average rate $\neq$ constant function — it just means endpoints match
Compare rates across intervals to find where the function changes fastest

(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

Determine if $f(x) = \frac{x^3}{x^2 + 1}$ is even, odd, or neither.

Step	Work
Compute $f(-x)$	$\frac{(-x)^3}{(-x)^2 + 1} = \frac{-x^3}{x^2 + 1}$
Compare to $-f(x)$	$-f(x) = \frac{-x^3}{x^2 + 1}$
$f(-x) = -f(x)$ ?	Yes → odd

If $f$ is odd and $f(3) = 7$ , find $f(-3) + f(3)$ .

Step	Work
Odd → $f(-3) = -f(3)$	$f(-3) = -7$
Sum	$-7 + 7 = 0$

Key insight: For any odd function, $f(-x) + f(x) = 0$ always. Also $f(0) = 0$ for any odd function (if $0$ is in the domain).

Symmetry and Graphs

How Even/Odd Shows on Graphs

Type	Symmetry	Test
Even	Fold along y-axis → halves match	Replace $x$ with $-x$ ; if same equation → even
Odd	Rotate 180° around origin → same graph	Replace $x$ with $-x$ ; if negated → odd

Worked Example 3

A graph passes through $(-2, 4)$ , $(0, 0)$ , and $(2, -4)$ . Could it be even or odd?

Check	Result
Even: $(-2, 4)$ and $(2, 4)$ ?	No — we have $(2, -4)$ , not

Products and Compositions

Operation	Even × Even	Odd × Odd	Even × Odd
Result	Even	Even	Odd

Example: $x^2 \cdot x^3 = x^5$ → even × odd = odd ✓

Even, Odd, or Neither? 🎯

Classify Each Function 🔍

Is each function even, odd, or neither?

Key Takeaways — Part 6

Property	Definition	Symmetry	Quick Test
Even	$f(-x) = f(x)$	y-axis	All terms have even exponents
Odd	$f(-x) = -f(x)$	Origin	All terms have odd exponents
Neither	Neither condition holds	No symmetry	Mixed exponents

Odd functions always pass through the origin (if defined at $x = 0$ )
Even × Even = Even; Odd × Odd = Even; Even × Odd = Odd
Most real functions are neither even nor odd

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 input vs. output

If $f(x) = ax + b$ , $f(2) = 7$ , and $f(5) = 16$ , find $a$ and $b$ .

Step	Work
Set up system	$2a + b = 7$ and $5a + b = 16$
Subtract	$3a = 9$ → $a = 3$
Find $b$	$6 + b = 7$ → $b = 1$
Answer	$f(x) = 3x + 1$

Find the range of $f(x) = -2(x - 3)^2 + 8$ .

Step	Work
Identify vertex	$(3, 8)$
Direction	Opens down ( $a = -2 < 0$ )
Maximum value	$8$ (at $x = 3$ )
Range	$(-\infty, 8]$ or $f(x) \leq 8$

Mixed Review 🎯

Hard SAT Function Patterns

Pattern 1: Nested Function Evaluation

$f(x) = x^2 - 1$ . What is $f(f(2))$ ?

$f(2) = 3$ , then $f(3) = 8$ . Answer: $8$ .

Pattern 2: Functions Defined by Conditions

" $f(x)$ is a linear function where $f(3) = 10$ and the rate of change is $-2$ ."

→ Slope $= -2$ : $f(x) = -2x + b$ . $f(3) = -6 + b = 10$ → . So .

Worked Example 3

$f$ is a quadratic with vertex $(1, -4)$ that passes through $(3, 0)$ . Find $f(x)$ .

Step	Work
Vertex form	$f(x) = a(x - 1)^2 - 4$
Plug in $(3,$

Worked Example 4

If $f(x) = 3x - 5$ , for what value of $x$ does $f(2x) = f(x) + 10$ ?

Step	Work
Expand $f(2x)$	$3(2x) - 5 = 6x - 5$
Expand

SAT-Level Challenge 🎯

Quick-Fire Function Review 🔍

Match each situation with the correct answer.

Key Takeaways — Full Functions & Graphs Review

Topic	One-Liner
Notation	$f(a)$ = plug in; $f(x) = k$ = solve
Composition	Inside out; order matters
Inverse	Swap $x$ / $y$ ; reflect over $y = x$
Transformations	Outside = vertical; inside = horizontal (opposite)
Piecewise	Check which rule applies at each $x$
Absolute value	V-shape; $
Rate of change	$\frac{f(b)-f(a)}{b-a}$ = secant slope
Even/Odd	Even → y-axis; Odd → origin
Domain	No ÷ 0, no $\sqrt{\text{negative}}$

Final tip: On the SAT, always check whether the question asks for an $x$ -value or a $y$ -value. "At what $x$ ..." vs. "What is the value of $f$ ..." are different questions!

∣ x - 4∣ \leq 3

Functions

Functions - Complete Interactive Lesson

Part 1: Function Notation

Functions & Graphs

What is a Function?

Domain & Range

SAT Function Notation Tricks

Worked Example 1

Worked Example 2

Function Evaluation with Tables

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

Worked Example 3 — "Solving" f(x)=kf(x) = kf(x)=k

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

Worked Example 1

Worked Example 2

Graph of Inverse

Composition with Tables

Verifying Inverses

Key Takeaways — Part 2

Horizontal Transformations (Inside the function)

Key Insight

Worked Example 1

Combining Multiple Transformations

Worked Example 2

Worked Example 3

Key Takeaways — Part 3

Absolute Value as Piecewise

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

Worked Example 1

SAT Graph Reading

Solving Absolute Value Equations

Worked Example 2

Worked Example 3

Key Takeaways — Part 4

Rate of Change

Worked Example 1

Intercepts

Comparing Rates of Change

Worked Example 2

Positive, Negative, and Zero

Key Takeaways — Part 5

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

Testing Algebraically

Neither Even nor Odd

Worked Example 1

Worked Example 2

Symmetry and Graphs

How Even/Odd Shows on Graphs

Worked Example 3

Products and Compositions

Key Takeaways — Part 6

SAT Strategies for Function Questions

Worked Example 1

Worked Example 2

Hard SAT Function Patterns

Pattern 1: Nested Function Evaluation

Pattern 2: Functions Defined by Conditions

Worked Example 3

Worked Example 4

Key Takeaways — Full Functions & Graphs Review

Composition: $f(g(x))$

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

Worked Example 3 — "Solving" $f(x) = k$

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

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

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