6.1 Function Notation, Evaluation, Domain, and Range
Key Takeaways
- AAF weights Functions at 10–20% of the 20-item CAT, typically 2–4 questions — the highest single Table 10 share.
- If f(x)=x^2+3x+1, then f(x+2)=(x+2)^2+3(x+2)+1=x^2+7x+11, which is not f(x)+2=x^2+3x+3.
- Polynomials have domain all real numbers; √(g(x)) needs g(x)≥0; a rational p(x)/q(x) needs q(x)≠0.
- Use function notation to add, subtract, combine, and evaluate — including f(x+2), which is not f(x)+2 — and name domain, range, max, and min.
- Inverse evaluation swaps coordinates on AAF: if f(3)=11, then f^{-1}(11)=3.
6.1 Function Notation, Evaluation, Domain, and Range
College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Functions at 10–20% of the 20-item computer-adaptive test — typically 2–4 items, the largest single share in Table 10. Table 11 names four skills: creating functions with function notation, evaluating linear and quadratic functions, graphing functions, and interpreting functions in a context. This section is the algebraic core: write f(x), evaluate it at numbers and at expressions such as x+2, and name domain and range.
What a function is, in notation
A function f assigns to each allowed input exactly one output. The input is the independent variable, usually x. The output is f(x), read “f of x,” not “f times x.” The rule may be a formula, a table, a graph, or a sentence. Creating a function with function notation means writing that rule as f(x) = … rather than as a loose y = …, so later items can ask for f(3), f(a), or f(x+2) without rewriting the story.
Linear and quadratic rules dominate AAF evaluation items:
| Type | Typical rule | Example |
|---|---|---|
| Polynomial (linear) | f(x) = mx + b | f(x) = 4x − 7 |
| Polynomial (quadratic) | f(x) = ax^2 + bx + c | f(x) = x^2 + 3x + 1 |
If f(x) = 4x − 7, then f(5) = 4(5) − 7 = 13 and f(−2) = 4(−2) − 7 = −15. Substitute the number, then simplify. Do not distribute a coefficient across a later addition that is not inside the function. If a table lists x = 3 next to f(x) = 11, then f(3) = 11 even when no formula is printed.
Evaluating f(a+2): substitution, not a translation of the output
The highest-yield trap on this skill is f(x+2) versus f(x)+2. f(x+2) means replace every x in the formula with the entire quantity (x+2). f(x)+2 means compute f(x) first, then add 2 to the output. Those are different functions unless f is a very special rule (for example a constant).
Worked example: f(x) = x^2 + 3x + 1, find f(x+2)
Let f(x) = x^2 + 3x + 1. Find f(x+2).
- Write the definition with a blank for the input:
f(□) = (□)^2 + 3(□) + 1. - Drop
x+2into every blank, with parentheses:f(x+2) = (x+2)^2 + 3(x+2) + 1. - Expand:
(x+2)^2 = x^2 + 4x + 4and3(x+2) = 3x + 6. - Combine:
x^2 + 4x + 4 + 3x + 6 + 1 = x^2 + 7x + 11.
So f(x+2) = x^2 + 7x + 11.
The wrong shortcut f(x)+2 = x^2 + 3x + 1 + 2 = x^2 + 3x + 3 is a different quadratic. Another trap is replacing only the squared x: (x+2)^2 + 3x + 1 = x^2 + 4x + 4 + 3x + 1 = x^2 + 7x + 5, which forgot to substitute into the linear term. Every x is an input slot.
Check with a number. If x = 1, then x+2 = 3 and f(3) = 9 + 9 + 1 = 19. The expanded form 1 + 7 + 11 = 19 matches. The false form f(1)+2 = 5 + 2 = 7 does not.
The same substitution works for a linear rule. If p(x) = 5x − 2, then p(a+2) = 5(a+2) − 2 = 5a + 10 − 2 = 5a + 8, which is not p(a)+2 = 5a. It is also not p(a+2) = 5a + 2. Parentheses around the whole input are not optional.
A related trap: f(3)+f(5) is not f(8) for a typical AAF function. If f(x) = 4x − 7, then f(3)+f(5) = 5 + 13 = 18, while f(8) = 25. Adding outputs is not the same as evaluating at the sum of the inputs.
Domain: which inputs are allowed
The domain of f is the set of inputs that the rule can legally accept. On AAF, “legally” means real-number outputs. Three families appear constantly; this section is a preview. Full solving of radical and rational equations, and the detailed graphs of those families, belong in the later radical/rational chapter (see Graphs, Domain, and Range of Radical and Rational Functions).
| Family | Typical form | Domain (real outputs) |
|---|---|---|
| Polynomial | x^2 + 3x + 1, 4x − 7 | All real numbers |
| Square root (even radical) | √(g(x)) | g(x) ≥ 0 |
| Rational | p(x)/q(x) | q(x) ≠ 0 |
Polynomials never divide and never take even roots of an expression in x, so nothing is forbidden. f(x) = x^2 + 3x + 1 has domain all real numbers.
A square root of a negative real number is not real, so h(x) = √(x − 4) needs x − 4 ≥ 0, hence x ≥ 4. The endpoint x = 4 is included because √0 = 0 is real. Cube roots are different: ∛x is defined for every real x, including negatives, so k(x) = ∛(x − 1) still has domain all reals.
A rational function is undefined where the denominator is zero. r(x) = (x+1)/(x−3) has domain all reals except x = 3. Factor first when the denominator is a quadratic: t(x) = 5/(x^2 − 9) is undefined at x = ±3 because x^2 − 9 = (x−3)(x+3).
When two restrictions appear together, keep both. u(x) = √(x+2)/(x−5) needs x+2 ≥ 0 and x ≠ 5, so x ≥ −2 except x = 5. Do not confuse domain with “where the graph is positive.” Domain is about allowed inputs, not the sign of the output.
Range: which outputs actually occur
The range is the set of output values f actually hits. For a nonconstant linear function, the range is all real numbers: f(x) = 4x − 7 runs from −∞ to ∞. A constant f(x) = 6 has range {6}.
For a quadratic that opens up, the range is all y at or above the vertex. f(x) = x^2 + 2 has vertex (0, 2) and range y ≥ 2. For a quadratic that opens down, the range is all y at or below the vertex. Finding that vertex is the same algebra as Quadratic Forms, Graphs, Vertex, and Intercepts: x = −b/(2a), then plug back in.
Even-powered polynomials that open down have a global maximum and a range of the form y ≤ M. Example: y = −2x^4 + 7 never exceeds 7, so the range is y ≤ 7. That idea is developed with graphs in Graphs of Functions, Vertical Line Test, Maxima and Minima.
From a table, the range is the set of listed outputs, and the domain is the set of listed inputs — unless the table is only a sample of a continuous rule given by a formula. AAF will ask you to name a maximum or minimum from a short table or a vertex, not just to recite the word “range.”
Combining and inverting (preview)
Beyond evaluation, AAF wants fluency with f(3)+g(3), (f−g)(x), and f(g(2)) when the formulas are short. If f(x) = x^2 and g(x) = x + 4, then f(2)+g(2) = 4 + 6 = 10 and (f−g)(2) = 4 − 6 = −2. Adding the formulas first, (f+g)(x) = x^2 + x + 4, then plugging in 2, gives the same 10.
Higher bands ask you to evaluate inverse functions. If a table says f(3) = 11, then f^{-1}(11) = 3. If f(x) = 2x − 5 and you need f^{-1}(9), solve 2x − 5 = 9 to get x = 7, so f^{-1}(9) = 7. You do not need the full inverse formula on every item, but you must know that the inverse undoes f and that f(f^{-1}(x)) = x on the range of f. The notation f^{-1} is not a reciprocal: 1/f(x) is a different function.
Handheld calculators are not allowed on AAF except with an approved accommodation (College Board calculator policy). Some items show an on-screen calculator icon; function-notation items usually do not. Substitution by hand is faster and avoids the f(x)+2 trap.
If f(x) = x^2 + 3x + 1, what is f(x+2)?
As a real-valued function, what is the domain of h(x) = √(x − 4)?
What is the range of f(x) = x^2 + 2 for real x?