Function Notation on the SAT
Evaluate and Compose with Confidence
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Function Notation
Function notation is compact, but it can feel confusing at first. The SAT uses it to test whether you can substitute carefully and follow the order of operations without getting lost in the symbols.
In this lesson, you will learn to treat a function like a machine, plug in values correctly, and handle compositions one step at a time. The goal is to make $f(x)$ feel as natural as any algebra expression.
A Simple Definition Unlocks Function Notation
A function is a rule that turns an input into an output. If $f(x) = 2x + 3$, then $f(4)$ means you replace $x$ with $4$ and compute $2(4) + 3$.
Composition means one function feeds into another. For $f(g(x))$, you find $g(x)$ first and then substitute that entire expression into $f$. Parentheses matter, so write the full substitution before simplifying.
Work Through Function Notation Step by Step
If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, what is $f(g(x))$?
Evaluate the composition step by step so the substitution stays clear.
Define $f(x)$ so you can evaluate it consistently.
Define $g(x)$ so you can compose it with $f$.
Substitute $g(x)$ into $f$ so the composition is explicit.
Simplify the expression to make the next step clear.
Use Desmos to Check Function Notation
If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, what is $f(g(x))$?
Desmos can evaluate functions and compositions quickly by defining functions and using them as inputs. This is helpful when the expressions are long.
f(x) = 2x + 3
f(g(x))
Algebra is usually faster for small expressions. Desmos is useful for checking or when the algebra is messy.
Desmos is faster when you need several function values or a quick check. Algebra is faster when the problem asks for a simplified expression.
Expert move: Define the function and evaluate it directly (for example,
f(3)), or use a table for multiple inputs; the graph shows domain restrictions
quickly.
When to skip Desmos: For a single short substitution, algebra is faster; use Desmos for long expressions or verification.
- Desmos features used: function definitions, function evaluation.
- Common mistake: treating $f(x + 1)$ as $f(x) + 1$.
Practice Function Notation with SAT-Style Questions
Evaluate each function carefully.
If , what is ?
If , what is ?
If and , what is ?
If , which expression equals ?
Key Takeaways to Remember for Function Notation
- Function notation means substitute carefully.
- Compositions follow the order $f(g(x))$.
- Desmos can verify compositions quickly.
About the Instructor
