A roller coaster climbs 200 feet in 30 seconds, then plummets the same distance in just 3 seconds. Same vertical change, wildly different experiences! Average rate of change is simply how much a function's output changes per unit of input — it's the slope of the line connecting two points on a graph. The ACT tests this concept in 3–4 questions, often using graphs, tables, or formulas. If you can compute slope between two points, you already know 80% of what you need. The other 20%? That's reading graph features — knowing where a function climbs, where it dips, and where it hits its peaks and valleys. Let's master both skills so you can grab every one of these points on test day.
A roller coaster climbs 200 feet in 30 seconds, then plummets the same distance in just 3 seconds. Same vertical change, wildly different experiences! Average rate of change is simply how much a function's output changes per unit of input — it's the slope of the line connecting two points on a graph. The ACT tests this concept in 3–4 questions, often using graphs, tables, or formulas. If you can compute slope between two points, you already know 80% of what you need. The other 20%? That's reading graph features — knowing where a function climbs, where it dips, and where it hits its peaks and valleys. Let's master both skills so you can grab every one of these points on test day.
The average rate of change of f(x) from x = a to x = b is:
(f(b) - f(a))/(b - a)
This is identical to the slope formula (y₂ - y₁)/(x₂ - x₁). You're computing the slope of the secant line — the straight line connecting the two points (a, f(a)) and (b, f(b)) on the graph.
Here's how to interpret the result:
- Positive rate: the function is generally increasing on that interval (output goes up as input goes up). - Negative rate: the function is generally decreasing (output drops as input rises). - Zero rate: the function ends where it started — though it may have moved up and down in between!
Think of it like a road trip. If you start at mile marker 10 and end at mile marker 70 after 2 hours, your average rate is 30 mph — even if you stopped for gas along the way.
A function is increasing on an interval where it goes up as you move from left to right. It's decreasing where it goes down as you move right. At a turning point (local max or min), the function switches from one behavior to the other.
The ACT asks questions like: "On what interval is f(x) increasing?" To answer, read the graph from left to right and identify the x-ranges where the curve goes up.
Here is the key language:
- "Increasing on (a, b)" means for every x in that interval, the function is rising. - "Decreasing on (c, d)" means it's falling throughout that interval. - We typically use open intervals (parentheses, not brackets) because the turning points themselves are neither increasing nor decreasing — they're momentary pauses.
For a standard upward-opening parabola like f(x) = x², the function decreases on (-∞, 0) and increases on (0, ∞). The vertex at x = 0 is where the switch happens.
Sixty percent of ACT test-takers miss at least one function composition question. That single mistake can drop your score by a full point. But composition is just plugging one function into another—once you see it that way, these problems become mechanical. Inverses are the reverse: if f turns 3 into 7, then f⁻¹ turns 7 back into 3. Think of a function as a machine and its inverse as the same machine running in reverse. This chapter makes both concepts automatic so you can grab those 3–5 questions per test with confidence.
f(g(x)) means: first apply g to x, then apply f to the result. Think of it as a two-step machine—x goes into g, the output goes into f.
The notation (f circ g)(x) means the same thing as f(g(x)). Read it right to left: g first, then f.
The most common mistake is multiplying the functions instead of composing them. Remember: f(g(x)) is NOT f(x) · g(x). You are substituting, not multiplying.
The inverse of f, written f⁻¹, reverses what f does. If f(3) = 7, then f⁻¹(7) = 3. The input and output swap roles.
Critical notation warning: f⁻¹(x) does NOT mean (1)/(f(x)). The -1 is not an exponent—it is the symbol for the inverse function. The ACT tests this distinction.
Not every function has an inverse. A function must be one-to-one (pass the horizontal line test) to have an inverse. Every output must trace back to exactly one input.
Imagine losing 3–4 easy points on the ACT just because function notation made you nervous. f(3) doesn't mean f times 3—it means "plug 3 into the function f." That's it. Whether the function is given as a formula, a table, or a graph, evaluation always means the same thing: find the output for a given input. Think of a function as an input-output machine. Whatever goes in the parentheses is your input. The formula, table, or graph tells you how to process it. What comes out is your output. These questions appear 3–4 times per test, and they follow predictable patterns every time.
When given f(x) = 2x² - 3x + 1, finding f(4) means replacing every x with 4:
f(4) = 2(4)² - 3(4) + 1 = 2(16) - 12 + 1 = 32 - 12 + 1 = 21
The golden rule: whatever is inside the parentheses replaces every instance of the variable in the formula. Always wrap the substituted expression in parentheses to avoid sign errors, especially with negative numbers and expressions.
When f is given as a table of values, f(3) means: find the row (or column) where x = 3 and read the corresponding f(x) value. Tables are the easiest representation—no calculation needed, just careful reading.
The ACT often provides tables with multiple functions (f, g, h) side by side, then asks composite questions like f(g(2)). Use the table twice: find g(2) first, then look up f of that result.
Every year, bacteria double their population while your ACT math score stays the same. Time to change that equation! Exponential and logarithmic functions appear in 2 to 3 questions per ACT, often nestled in the harder problems that separate good scores from great ones. These functions model everything from compound interest to radioactive decay, from population growth to pH levels. The secret weapon? Understanding that logarithms and exponentials are inverse operations — like multiplication and division on steroids. Master these relationships, and you'll unlock problems that stump 70 percent of test-takers. Today, you'll learn to recognize exponential patterns instantly, convert between forms effortlessly, and solve these problems in under 60 seconds.
Exponential functions pack more punch than any other function family on the ACT. The basic form y = ab^x tells a story: a is your starting point (the y-intercept when x = 0), b is your multiplier (the growth or decay factor), and x is typically time or some other variable in the exponent.
When b is greater than 1, you have exponential growth that starts slowly then explodes upward. When b is between 0 and 1, you have decay that plummets quickly then levels off. The ACT loves base 2 for doubling problems, base 10 for scientific notation, and occasionally base e for continuous growth. Remember this golden rule: exponential functions never go negative or hit zero — they just get infinitely close.
Every sixty seconds on the ACT math section, another student encounters a function question and freezes. But here's the secret the test makers don't advertise: function questions follow predictable patterns that, once recognized, turn intimidating problems into easy points. You'll face three to four function questions per test, covering notation, evaluation, domain, and range. Functions are simply mathematical machines that transform inputs into outputs following specific rules. Master these patterns, and you'll transform confusion into confidence. By the end of this lesson, you'll decode function notation like f(x), pinpoint domains and ranges instantly, and navigate even the trickiest composite functions. These aren't just abstract concepts — they're your keys to unlocking ten percent of the ACT math section.
Function notation might look like algebraic hieroglyphics at first, but it's actually a simple labeling system. When you see f(x), think of f as the function's name — like calling your friend Sam — and x as what you're feeding into the function. The notation f(x) = 2x + 3 tells you that function f takes any input, doubles it, then adds three.
Imagine dropping a coin into a vending machine. You select your item, the machine processes your choice, and out comes your snack. Functions work exactly the same way. To evaluate f(3) when f(x) = 2x² - 3x + 1, replace every x with 3 and simplify carefully using order of operations.
Two thirds of students miss piecewise function questions on the ACT, yet these problems follow predictable patterns that take just minutes to master. Piecewise and absolute value functions appear in 3 to 4 questions per test, often determining the difference between a 30 and a 34 in math. These functions work like traffic lights, with different rules applying in different zones. Piecewise functions change their formula based on x-values, while absolute value functions create that distinctive V shape by flipping negative outputs positive. Once you recognize their visual patterns and evaluation strategies, these intimidating problems transform into quick wins. Master these concepts now and watch your math score jump.
Piecewise functions operate like different speed limits on a highway. Each section has its own rule. The function uses different formulas for different x-intervals, written with a large bracket showing which formula applies when.
Absolute value functions measure distance, always giving non-negative results. The basic |x| creates a perfect V shape with its point at the origin. For positive x values, it acts like y = x. For negative x values, it flips the sign, acting like y = -x.
Imagine taking the square root of negative four. Your calculator throws an error, right? That single moment captures why radical functions are worth two to three questions on every ACT math section. These functions behave differently than polynomials or linear equations, creating unique challenges around domain, range, and transformations. Radical functions test whether you truly understand function behavior, not just formula memorization. Throughout this lesson, you'll master finding valid input values, determining output ranges, and navigating transformations. Most importantly, you'll learn to spot the tricky radical problems that trip up even strong math students. Success with radicals often means the difference between a 28 and a 32 on the math section.
Square roots hate negative numbers. This simple fact drives every domain question on the ACT. When you see √(x - 3), your first thought should be: when is x - 3 non-negative? Set x - 3 ≥ 0, solve for x ≥ 3, and you've found your domain.
Range questions test whether you understand how radical functions behave. Start with the parent function: √(x) has range [0, ∞) because square roots output non-negative values. Now watch transformations work their magic.
Sixty percent of ACT test-takers miss transformation questions not because they can't do algebra, but because they mix up which direction to shift. Linear function transformations appear in 3 to 5 questions per test, often disguised within word problems or coordinate geometry. These questions test whether you can predict how changes to a function's equation affect its graph. Master four key moves — shifting up or down, sliding left or right, stretching or compressing, and flipping across axes. Each transformation follows predictable rules that, once understood, turn these questions into quick points. By the end of this lesson, you'll decode any linear transformation in under 30 seconds.
Vertical shifts are the simplest transformation to master. When you add a constant k to any linear function, you lift every point on the line up by k units. Think of it as grabbing the entire line and sliding it straight up or down without tilting.
Horizontal shifts trip up more students than any other transformation because the signs work backwards from intuition. When you see f(x - 2), the graph shifts RIGHT 2 units, not left. Why? Think about what value of x gives you the original function's y-intercept. For f(x - 2) to equal f(0), you need x = 2. So the y-intercept point, originally at x = 0, now sits at x = 2. The entire graph slides right.
Imagine flipping a graph upside down with one negative sign or sending it soaring to infinity based on a single exponent. That power lies in understanding polynomial transformations and end behavior — topics that appear in 3 to 5 questions on every ACT Math section. These concepts reveal how functions move, stretch, and behave at their extremes. You'll master shifting graphs horizontally and vertically, stretching and reflecting them, and predicting where polynomials head as x approaches infinity. From basic parabolas to complex fifth-degree polynomials, you'll decode their movement patterns and long-term behavior. This skill transforms abstract algebra into visual mastery, turning potential confusion into confident problem-solving.
Every ACT tests your ability to slide functions around the coordinate plane. The trickiest part catches most students off guard: horizontal shifts work backwards from intuition.
Stretching and flipping graphs builds on shifts but adds multiplicative magic. When you multiply the entire function by a constant, you stretch or compress vertically. Multiply by 3, and every y-value triples. Multiply by (1)/(2), and the graph shrinks to half its height. The critical move: multiply by -1, and the entire graph flips over the x-axis like a pancake.