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Function notation is a compact way to name a rule and the input sent through it. On the SAT, most notation questions become routine once you separate the input, the rule, and the resulting output.
In f(x), f names the function and x is its input. The expression f(a) means use a wherever the rule uses its input, while f(a)=b says that input a produces output b.
Key takeaways
- f(a) is an output request, not multiplication of f and a.
- In a table or graph, f(a)=b corresponds to the point (a,b).
- For f(g(2)), find the inner output g(2) first, then use that value as the input to f.
- f(x)+k shifts outputs vertically, while f(x+k) changes inputs and shifts the graph horizontally in the opposite visual direction.
What does function notation mean on the SAT?
A function is a rule that assigns exactly one output to each allowed input. The rule can appear as an equation, table, graph, or verbal instruction. The notation does not require a familiar formula. If a table says the input 3 is paired with the output 11, then f(3)=11 even when no algebraic rule is given.
| Notation | Instruction | Representation |
|---|---|---|
| f(4) | Find the output when the input is 4 | Substitute 4, use the row x=4, or read the graph at x=4 |
| f(a)=b | Input a produces output b | The graph contains (a,b) |
| f(x)=g(x) | Find inputs where outputs match | Look for intersections |
| f(g(2)) | Apply g, then apply f | Work from the inside outward |
The SAT linear functions guide explains slope, intercepts, and linear models. The SAT nonlinear equations guide covers quadratics and nonlinear systems. This child focuses on the notation shared by those families, whether the rule is linear, nonlinear, tabular, graphical, or simply stated in words.
How do you evaluate $f(a)$?
- 1
Mark the input
In f(-3), the entire input is -3. Keep its parentheses.
- 2
Replace every input variable
If f(x)=2x^2-x, write f(-3)=2(-3)^2-(-3).
- 3
Follow the operation order
Square before multiplying, then combine the terms.
- 4
Check the requested output
Confirm that the stem asks for f(a) rather than the input that makes f(x) equal a given value.
When the input is negative, substitute it with parentheses. For x^2, the value (-3)^2 is 9. Writing -3^2 changes the expression before the arithmetic has started.
Tables and graphs use the same idea without substitution. To find f(5) in a table, locate the row whose input is 5 and read its output. On a graph, move to x=5 and read the corresponding y-coordinate. To solve f(x)=5, reverse the search: find the input or inputs whose output is 5.
How do simple composite functions work?
Composition sends one function's output into another function. For f(g(2)), calculate g(2) first. If g(2)=7, the remaining task is f(7). The function names show the order, and the nested parentheses provide a small set of directions with no interest in your preferred reading direction.
Wrong order
Apply f to 2 because f appears first on the page.
Inside first
Evaluate the innermost expression g(2), then place that output into f.
What do $f(x)+k$ and $f(x+k)$ change?
f(x)+k adds k after the function acts, so every output rises by k when k>0. The graph moves up. By contrast, f(x+k) changes the input before the function acts. The old output at input a now occurs when x+k=a, or x=a-k, so the graph moves left by k when k>0.
Keep the distinction conceptual: outside changes affect outputs, while inside changes affect inputs. The broader SAT Math strategies guide shows when to use substitution, a table, algebra, or a graph after you translate what the notation requests.
Can you evaluate one function?
The function f is defined by f(x)=2x^2-3. What is the value of f(-2)?
Translate the requested input and output first, then choose substitution, a table, or a graph.
Practice function notation →Frequently asked questions
- What does f of x mean on the SAT?
- It names the output produced by function f when the input is x. The letter in parentheses is an input, not a multiplication factor.
- How do you evaluate a function at a number?
- Replace every input variable in the rule with that number, keep negative inputs in parentheses, and follow the usual order of operations.
- How do you read a function from a table or graph?
- Find the given input and read its paired output. On a graph, the input is the x-coordinate and the function value is the y-coordinate.
- What does a composite function mean?
- A composite function uses one function's output as another function's input. Evaluate the innermost function first and then work outward.