Nonlinear Equations and Functions

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 $(a + 1)$"

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 $f(a + 2)$.

Worked Example 2

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

Function Evaluation 🎯

Function Evaluation with Tables

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

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 $f(x) = 0$?

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

Harder Function Notation 🎯

What Does the Notation Mean? 🔍

For each expression, choose what it represents.

Key Takeaways — Part 1

• 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

Functions & Graphs

Part 2 of 7 — Composition and Combining Functions

Composition: $f(g(x))$

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

Example: $f(x) = x^2$ and $g(x) = x + 3$

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

$g(f(2)) = g(4) = 7$ — order matters!

Substituting an Expression: $f(x + 1)$, $f(2x)$

$f(\text{anything})$ means "replace every $x$ in the rule with that anything, in parentheses."

• If $f(x) = 3x - 4$, then $f(2x) = 3(2x) - 4 = 6x - 4$
• $f(x + 1)$ is not the same as $f(x) + 1$: the first changes the input, the second changes the output

Combining Functions

Worked Example 1

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

Worked Example 2

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

Composition & Combining Functions 🎯

Composition with Tables

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

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

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

Working a Composition Backward

Sometimes the SAT gives the value of a composition and asks for the input.

Example: $f(x) = 2x + 3$ and $g(x) = x^2$. If $f(g(k)) = 21$ and $k > 0$, find $k$.

Harder Composition 🎯

Composition Order Matters! 🔍

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

Key Takeaways — Part 2

Functions & Graphs

Part 3 of 7 — Transformations of Functions

Vertical Transformations (Outside the function)

Horizontal Transformations (Inside the function)

Key Insight

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

Worked Example 1

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

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

Transformations 🎯

Combining Multiple Transformations

Apply transformations in this order:

1. Horizontal shifts and stretches (inside)
2. Reflections
3. Vertical stretches (outside)
4. Vertical shifts (outside)

Worked Example 2

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

Worked Example 3

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

Applied Transformations 🎯

Name That Transformation 🔍

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

Key Takeaways — Part 3

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

Functions & Graphs

Part 4 of 7 — Piecewise & Absolute Value Functions

Piecewise Functions

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

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

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

Worked Example 1

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

SAT Graph Reading

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 $|2x - 5| = 7$.

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

Worked Example 3

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

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

Absolute Value Equations 🎯

Piecewise or Absolute Value? 🔍

Classify each function type and identify key features.

Key Takeaways — Part 4

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:

• "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)

Rate of Change

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.

Worked Example 1

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

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

Intercepts

• 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:

Where is $f$ increasing fastest?

$f$ 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

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

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

Functions & Graphs

Part 6 of 7 — Exponential Functions and Their Graphs

The Form $f(x) = a \cdot b^x$

• $a$ is the initial value: $f(0) = a \cdot b^0 = a$, so the $y$-intercept is $(0, a)$
• $b$ is the growth factor: each time $x$ goes up by 1, the output is multiplied by $b$
• $b > 1$ → growth; $0 < b < 1$ → decay

Reading the Percent from $b$

Linear vs. Exponential

• Linear: the output adds the same amount each step (constant difference)
• Exponential: the output is multiplied by the same factor each step (constant ratio)

Worked Example 1

A function $f$ has $f(0) = 5$, $f(1) = 15$, $f(2) = 45$, $f(3) = 135$. Write $f(x)$.

Worked Example 2

For $f(x) = 800(0.75)^x$, describe the function and find $f(2)$.

Exponential Basics 🎯

Features of Exponential Graphs

For $f(x) = a \cdot b^x + k$ with $a > 0$:

Worked Example 3

Find the intercepts of the graph of $y = 3(2)^x - 12$.

Worked Example 4 — Other Time Units

A quantity starts at 40 and doubles every 3 years: $Q(t) = 40(2)^{t/3}$. Find $Q(9)$.

$9$ years is $9/3 = 3$ doubling periods: $Q(9) = 40(2)^3 = 320$.

Key insight: In $b^{t/n}$, the output is multiplied by $b$ once every $n$ units of $t$.

Comparing Growth

$g(x) = 100 + 20x$ (linear) and $h(x) = 100(1.2)^x$ (exponential) both start at 100 and both equal 120 at $x = 1$. At $x = 5$: $g(5) = 200$ but $h(5) \approx 248.8$. An increasing exponential eventually passes any linear function.

Exponential Graphs & Models 🎯

Read the Exponential 🔍

Choose the correct description for each function.

Key Takeaways — Part 6

• An increasing exponential function eventually exceeds any increasing linear function

Functions & Graphs

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

Functions Cheat Sheet

SAT Strategies for Function Questions

1. Use the answer choices — if asked for a function and given formulas, test with a value
2. Read graphs carefully — pay attention to open vs. closed circles
3. Don't confuse $f(a) = b$ with $f(b) = a$ — in the first, $a$ is the input; in the second, $a$ is the output
4. For word problems: identify input vs. output

Worked Example 1

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

Worked Example 2

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

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 $= -2$: $f(x) = -2x + b$. $f(3) = -6 + b = 10$ → $b = 16$. So $f(x) = -2x + 16$.

Worked Example 3

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

Worked Example 4

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

SAT-Level Challenge 🎯

Quick-Fire Function Review 🔍

Match each situation with the correct answer.

Key Takeaways — Full Functions & Graphs Review

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!