2.1 Function Notation and Representations
Key Takeaways
- On Praxis 5165, f(x) denotes the output of rule f at input x; domain is all valid inputs and range is all resulting outputs.
- A table with constant ratios between consecutive outputs models exponential growth; constant first differences model linear growth.
- Function composition (f ∘ g)(x) means evaluate g first, then apply f to that result — order matters.
- The vertical line test checks whether a graph represents a function; the horizontal line test checks whether a function is one-to-one and has an inverse.
- To evaluate f⁻¹(k), solve f(x) = k for x rather than reciprocating the expression 1/f(x).
Why This Section Matters
Functions and Calculus is a 30% domain on Praxis Mathematics Content Knowledge (5165) — tied with Number and Quantity and Algebra for the largest share of scored content. ETS repeatedly tests whether you can move fluently among symbolic, graphical, tabular, and verbal representations of the same relationship. Many items look like routine algebra until you notice function structure: composition, inverses, domain restrictions, or the difference between a relation and a function.
Because roughly 25% of Praxis 5165 questions embed mathematics inside task-of-teaching scenarios, you may also be asked which representation best reveals a pattern, or how to respond when a student confuses inverse notation with reciprocals. Solid function vocabulary is the foundation for every later topic in this chapter.
Function Notation and Core Vocabulary
A function assigns exactly one output to each input in its domain. We write f(x) to mean "the output produced by rule f when the input is x." The set of allowable inputs is the domain; the set of resulting outputs is the range.
| Symbol | Meaning | Quick Example |
|---|---|---|
| f(3) | Evaluate f at x = 3 | If f(x) = 2x + 1, then f(3) = 7 |
| f⁻¹(x) | Inverse-function output | The input x that makes f return a given value |
| (f ∘ g)(x) | Composition f(g(x)) | Apply g first, then f |
Piecewise functions use different rules on different intervals. For example, a shipping fee might be C(w) = 5 when w ≤ 2 pounds and C(w) = 5 + 1.5(w - 2) when w > 2. On Praxis items, identify which rule applies before substituting.
Domain and Range
Domain restrictions come from context and from algebra:
- Square roots require a nonnegative radicand: f(x) = √(x - 4) needs x ≥ 4.
- Rational functions exclude zeros of the denominator: h(x) = 1/(x - 2) requires x ≠ 2.
- Even roots of even powers in the denominator create additional restrictions.
For f(x) = √(x - 4), the range is y ≥ 0 because square-root outputs are never negative in the real-number setting tested on Praxis 5165.
Multiple Representations on the Exam
ETS presents the same relationship in different forms and expects you to translate among them:
| Representation | What to Look For |
|---|---|
| Equation | Structure (linear, quadratic, exponential) |
| Table | Constant difference → linear; constant ratio → exponential |
| Graph | Shape, intercepts, asymptotes, holes |
| Verbal | "Twice the input plus five" → f(x) = 2x + 5 |
Worked Example: Exponential vs. Linear from a Table
A teacher shows x: 0, 1, 2, 3 and y: 5, 10, 20, 40.
- First differences: 5, 10, 20 — not constant, so the pattern is not linear.
- Ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2 — constant multiplier 2.
The model is y = 5(2^x). A strong instructional response highlights multiplicative change as the signature of exponential growth.
Worked Example: Building an Equation from a Verbal Rule
"A function outputs 3 more than twice its input." Translate directly: f(x) = 2x + 3. If the stem adds "when the input is negative, output is always 1," you now have a piecewise rule requiring separate cases.
Composition and Inverses
Composition nests one function inside another: (f ∘ g)(x) = f(g(x)). The inside function runs first.
Worked Example: Composition Order
Let f(x) = x² + 1 and g(x) = 2x - 3. Find (f ∘ g)(2).
- g(2) = 2(2) - 3 = 1
- f(1) = 1² + 1 = 2
So (f ∘ g)(2) = 2. Reversing order gives (g ∘ f)(2) = g(5) = 7 — composition is not commutative.
Worked Example: Inverse Evaluation
If f(x) = 3x - 5, what is f⁻¹(7)?
Solve 3x - 5 = 7 → 3x = 12 → x = 4. Therefore f⁻¹(7) = 4.
Check: f(4) = 12 - 5 = 7. Notice that f⁻¹(7) is not 1/f(7) and not f(7)⁻¹.
To find an inverse rule symbolically, swap x and y, then solve for y. For f(x) = 3x - 5, swap to x = 3y - 5, so y = (x + 5)/3.
Graphical Tests
| Test | Question | Pass Means |
|---|---|---|
| Vertical line test | Is the graph a function? | No vertical line intersects more than once |
| Horizontal line test | Is the function one-to-one? | No horizontal line intersects more than once |
A parabola opening upward passes the vertical line test (it is a function) but fails the horizontal line test (it is not one-to-one on all reals), so it lacks a global inverse unless the domain is restricted — for example to x ≥ 0.
Connecting to Praxis 5165 Logistics
The computer-delivered exam provides an on-screen graphing calculator. Use it to verify graphs after you have set up the function relationship symbolically. Calculator exploration supports teaching items about which representation (table, graph, or equation) best helps students see a pattern.
Common Praxis Traps
- Writing f⁻¹(x) = 1/f(x) — inverse notation is not a reciprocal.
- Applying (f ∘ g) as f(x) · g(x) — composition is not multiplication.
- Ignoring domain restrictions when simplifying rational expressions (covered more in Section 2.3).
- Claiming the vertical line test checks invertibility — that is the horizontal line test.
Section Takeaways
Read f(a) as substitution into the rule, build inverses by reversing operations, and practice switching among tables, graphs, equations, and words. Pair this section with function-notation items on /practice/praxis-math before moving to transformations.
If f(x) = 3x - 5, what is f⁻¹(7)?
Given f(x) = x² + 1 and g(x) = 2x - 3, what is (f ∘ g)(2)?
A table shows x: 0, 1, 2, 3 and y: 5, 10, 20, 40. Which observation best identifies an exponential pattern?