Last updated July 07, 2026 · Reviewed by the AnvayaPrep team
Introduction
Functions and Nonlinear Models is the unit covering function concepts and all nonlinear function types tested on the SAT, from foundational function evaluation through model selection and applied word problems. The unit's 20 topics span function fundamentals (domain, range, function evaluation, function tables, inverse functions, composite functions, zeros of functions, maximum and minimum), nonlinear function types (quadratic functions, exponential functions, absolute value functions, piecewise functions), graphical analysis (interpreting graphs, nonlinear graphs, average rate of change, comparing functions), transformations (function transformations), and applied contexts (model selection, nonlinear word problems, SAT function traps).
Learning Objectives
- Evaluate a function at a specific input by substituting the value for the variable and simplifying
- Determine the domain (set of valid inputs) and range (set of possible outputs) of a function from its equation or graph
- Evaluate composite functions f(g(x)) by working from the inside out: compute g(x) first, then use that result as input for f
- Find the inverse of a function by switching x and y and solving for y; verify by confirming that f(f^(-1)(x)) = x
- Identify and apply the four function transformations: vertical shifts f(x) + k, horizontal shifts f(x - h), vertical stretches a*f(x), and reflections -f(x) and f(-x)
- Read key features from graphs: zeros (x-intercepts), maximum and minimum values, average rate of change over an interval
- Distinguish between linear, quadratic, and exponential models using first differences (linear), second differences (quadratic), and constant ratios (exponential)
- Set up and solve nonlinear word problems by identifying the appropriate function type, writing the model, and solving for the target quantity
High-Yield Concepts
| Concept | What It Tests | Key Rule |
|---|---|---|
| Function evaluation | Substituting inputs into functions | Replace every instance of the variable with the given value, then simplify |
| Composite functions | Chaining two functions together | f(g(x)): apply g first, then apply f to the result |
| Domain | Valid inputs for a function | Exclude values that make denominators zero or produce negatives under even radicals |
| Inverse functions | Reversing input-output roles | Swap x and y, solve for y; the result is f^(-1)(x) |
| Vertical translation | Shifting a graph up or down | f(x) + k shifts up k units; f(x) - k shifts down k units |
| Horizontal translation | Shifting a graph left or right | f(x - h) shifts right h units; f(x + h) shifts left h units (counterintuitive direction) |
| Reflection | Flipping a graph over an axis | -f(x) reflects over x-axis; f(-x) reflects over y-axis |
| Model selection | Choosing the right function type | Constant first differences: linear; constant second differences: quadratic; constant ratios: exponential |
| Average rate of change | Measuring change over an interval | (f(b) - f(a)) / (b - a) for the interval [a, b] |
| Zeros of functions | Finding where f(x) = 0 | Set the function equal to zero and solve; zeros are x-intercepts on the graph |
Study Strategy
Start with function evaluation, since all other topics in this unit depend on it. Practice substituting numbers, expressions, and other functions as inputs. The notation f(3) means "evaluate f when the input is 3." The notation f(g(x)) means "evaluate f when the input is g(x)."
Study domain and range together. Domain is about valid inputs (look for denominators that could be zero, or even-index radicals that require the radicand to be non-negative). Range is about possible outputs (look at the graph or analyze the function's behavior).
Master composite functions by working strictly inside-out: always evaluate the inner function first, then use that result as the input for the outer function. The order matters: f(g(x)) is generally different from g(f(x)).
Learn function transformations as a systematic table. Transformations outside the function (added to or multiplying f(x)) affect y-values and are intuitive: adding k moves the graph up, multiplying by a stretches it. Transformations inside the function (modifying the input x) affect x-values and are counterintuitive: replacing x with (x - h) moves the graph right, not left.
Study model selection by learning to distinguish the three main types from a table of values. Constant first differences (y increases by the same amount each step): linear. Constant second differences: quadratic. Constant ratios between consecutive y-values: exponential. In word problems, identify the type from context: compound growth is exponential, area or projectile problems are quadratic, constant-rate problems are linear.
Common Mistakes
Confusing the direction of horizontal translations. The function f(x - 3) shifts the graph 3 units to the RIGHT, not left. Students who read "minus 3" as "move left" make systematic errors on transformation questions. The rule is: f(x - h) shifts right h units because to get the same y-value, you need x to be h units larger.
Reversing the order in composite functions. In f(g(2)), you evaluate g(2) first, then use that number as the input for f. Students who evaluate f first and use f(2) as the input for g have reversed the order. The inner function is always evaluated first.
Finding the inverse by negating rather than solving. The inverse of f(x) = 2x + 3 is NOT f^(-1)(x) = -2x - 3. To find the inverse, replace f(x) with y, swap x and y to get x = 2y + 3, then solve for y: y = (x - 3)/2. The inverse undoes the original function, not just negates it.
Using first differences when the model is exponential. If first differences are not constant, students sometimes guess "quadratic" without checking whether ratios are constant. Always check ratios (divide consecutive y-values) when first differences are not constant. Constant ratios indicate exponential, not quadratic.
Misreading zeros and maximums from graphs. Zeros are where the graph crosses the x-axis (not the y-axis). Maximums and minimums are the highest/lowest points, identified by y-value (not x-value). The SAT asks for specific coordinates; always read the correct axis.
Exam Tips
For composite function problems, write out the intermediate step: compute g(2) explicitly, write down the number, then compute f(that number). Do not try to track two substitutions mentally in one step. The extra two seconds spent writing the intermediate value prevents errors that lose points.
On model selection questions where the SAT gives a table and asks which equation matches, check the rate-of-change pattern before doing any algebra. If first differences are constant, eliminate quadratic and exponential options immediately. If ratios are constant, eliminate linear and quadratic. Only after identifying the type should you check coefficients to pick the specific equation.
To remember transformation direction for inputs versus outputs: transformations that appear OUTSIDE the function follow their sign directly (f(x) + 3 moves the graph UP 3). Transformations that appear INSIDE the function go in the OPPOSITE direction (f(x - 3) moves the graph RIGHT, not left). Outside = obvious direction, inside = opposite direction.
Average rate of change over an interval [a, b] is (f(b) - f(a)) / (b - a), which is the slope of the secant line connecting the two points on the graph. For linear functions, this equals the constant slope for any interval. For nonlinear functions, it varies by interval. SAT questions often present a graph and ask for the average rate of change over a specific interval, requiring you to read the function values at the endpoints.
Sign up free to keep reading
Create a free AnvayaPrep account to finish this SAT guide on Functions and Nonlinear Models — plus flashcards and practice questions.