Functions (SAT Math)

What is a Function?

A function is a relationship where each input has exactly ONE output.

Think of it like a machine:

• You put in a number (input $x$)
• The function processes it
• You get a result (output $y$ or $f(x)$)

Function Notation

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

• Function name: $f$
• Input variable: $x$
• Rule: Multiply input by 2, then add 3

Example evaluations:

• $f(5) = 2(5) + 3 = 13$
• $f(0) = 2(0) + 3 = 3$
• $f(-2) = 2(-2) + 3 = -1$

Domain and Range

Domain (inputs)

All possible $x$-values that can be used in a function

Common restrictions:

• Cannot divide by zero
• Cannot take square root of negative (in real numbers)
• Explicit restrictions stated in problem

Example 1: $f(x) = \frac{1}{x-3}$

Domain: All real numbers EXCEPT $x = 3$ (would make denominator zero)

Example 2: $f(x) = \sqrt{x + 5}$

Domain: $x + 5 \geq 0$, so $x \geq -5$

Range (outputs)

All possible $y$-values the function can produce

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

• Domain: All real numbers
• Range: $y \geq 0$ (squares are never negative)

Example: $f(x) = -x^2 + 4$

• Domain: All real numbers
• Range: $y \leq 4$ (parabola opens down, vertex at $y = 4$)

Evaluating Functions

Direct Substitution

Given: $f(x) = x^2 - 3x + 2$, find $f(4)$

Solution: $f(4) = (4)^2 - 3(4) + 2 = 16 - 12 + 2 = 6$

Substituting Expressions

Given: $f(x) = x^2 - 3x + 2$, find $f(a + 1)$

Solution: $f(a+1) = (a+1)^2 - 3(a+1) + 2$ $= a^2 + 2a + 1 - 3a - 3 + 2$ $= a^2 - a$

Key: Replace EVERY $x$ with $(a+1)$

Composite Functions

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

"$f$ composed with $g$" means: do $g$ first, then do $f$ to the result

Example

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

Find $f(g(3))$:

Step 1: Find $g(3)$
$g(3) = 3^2 = 9$

Step 2: Find $f(9)$
$f(9) = 2(9) + 1 = 19$

Therefore: $f(g(3)) = 19$

Order Matters!

Find $g(f(3))$:

Step 1: Find $f(3)$
$f(3) = 2(3) + 1 = 7$

Step 2: Find $g(7)$
$g(7) = 7^2 = 49$

Therefore: $g(f(3)) = 49 \neq f(g(3))$

Important: $f(g(x)) \neq g(f(x))$ in general!

Interpreting Function Graphs

Reading Values from Graphs

To find $f(3)$:

1. Find $x = 3$ on horizontal axis
2. Go up/down to the curve
3. Read the $y$-value

To find when $f(x) = 5$:

1. Find $y = 5$ on vertical axis
2. Go left/right to the curve
3. Read all $x$-values where curve crosses $y = 5$

Key Features

x-intercepts (zeros): Where graph crosses x-axis
→ $f(x) = 0$

y-intercept: Where graph crosses y-axis
→ $f(0)$

Maximum: Highest point (vertex of downward parabola)

Minimum: Lowest point (vertex of upward parabola)

Increasing: Graph goes up as you move right
Decreasing: Graph goes down as you move right

Function Transformations

Vertical Shifts

$f(x) + k$: Shift UP by $k$ units
$f(x) - k$: Shift DOWN by $k$ units

Example: If $f(x) = x^2$, then:

• $f(x) + 3 = x^2 + 3$ shifts parabola up 3 units

Horizontal Shifts

$f(x - h)$: Shift RIGHT by $h$ units (opposite of what you'd think!)
$f(x + h)$: Shift LEFT by $h$ units

Example: If $f(x) = x^2$, then:

• $f(x - 2) = (x-2)^2$ shifts parabola right 2 units

Reflections

$-f(x)$: Reflect over x-axis (flip upside down)
$f(-x)$: Reflect over y-axis (flip left-right)

Stretches

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

Word Problems with Functions

Example: Cost Function

A gym charges a $50 membership fee plus$30 per month.

Function: $C(m) = 50 + 30m$

• $C$ = total cost
• $m$ = number of months

Questions:

• $C(6) = 50 + 30(6) = 230$ → Cost for 6 months
• If $C(m) = 200$, solve: $50 + 30m = 200$ → $m = 5$ months

Example: Distance Function

A car travels at 60 mph for $t$ hours.

Function: $d(t) = 60t$

Questions:

• $d(3) = 60(3) = 180$ → Distance after 3 hours
• If $d(t) = 240$, solve: $60t = 240$ → $t = 4$ hours

SAT Question Types

Type 1: Evaluate $f(a)$

Strategy: Substitute $a$ for every $x$ and simplify

Type 2: Solve $f(x) = k$

Strategy: Set function equal to $k$ and solve for $x$

Example: If $f(x) = 2x - 7$ and $f(x) = 11$: $2x - 7 = 11$ $2x = 18$ $x = 9$

Type 3: Find $f(g(x))$ or $g(f(x))$

Strategy: Work from inside out

Type 4: Interpret Graphs

Strategy:

• Trace with your finger
• Check $x$-value → $y$-value
• Verify answer makes sense

Type 5: Domain/Range from Graph or Equation

Strategy:

• Domain: Look at $x$-values covered
• Range: Look at $y$-values achieved
• Check for restrictions (division by zero, square roots)

Common Mistakes

❌ Confusing $f(x+2)$ with $f(x) + 2$

• $f(x+2)$ shifts graph LEFT 2
• $f(x) + 2$ shifts graph UP 2

❌ Wrong order in composition

• $f(g(x))$ means do $g$ FIRST
• Not the same as $g(f(x))$!

❌ Not using parentheses

• $f(3+1) = f(4)$, not $f(3) + 1$

❌ Forgetting domain restrictions

• $\frac{1}{x}$ has no value at $x = 0$

❌ Misreading graphs

• Check which axis is which
• Verify scale (not always by 1's)

Quick Tips for SAT

✓ Function notation is just substitution — replace $x$ with whatever is in parentheses
✓ Graphs tell you everything — use them to find values quickly
✓ Order matters in composition — inside function first, outside function second
✓ Domain = possible inputs → check what $x$ CAN'T be
✓ Range = possible outputs → check what $y$ values are achieved
✓ Transformations stack — multiple shifts/stretches apply in sequence

Practice Approach

1. Identify function type (linear, quadratic, etc.)
2. Check what's being asked (evaluate, solve, compose, transform)
3. Use appropriate strategy
4. Double-check your substitution (most common error)
5. Verify answer makes sense (domain/range, reasonableness)