4.1 Function Notation, Domain, Range & Evaluation

4.1 Function Notation, Domain, Range & Evaluation

Functions serve as the central organizational language of modern mathematics. On the CLEP College Mathematics exam, candidates must fluidly interpret mathematical relations, manipulate function notation, evaluate algebraic expressions, identify restricted domains, and analyze piecewise-defined rules.

---

1. Relations, Functions & The Vertical Line Test

A relation is any set of ordered pairs $(x, y)$ establishing a relationship between an input set and an output set.

• Domain: The set of all valid independent input values ($x$-values).
• Range: The set of all resulting dependent output values ($y$-values or $f(x)$).

The Formal Definition of a Function

A function is a specialized relation in which each element in the domain corresponds to exactly one element in the range. While multiple distinct inputs may yield the identical output (e.g., $(-2)^2 = 4$ and $2^2 = 4$), a single input can never produce two different outputs.

+-----------------------------------------------------------------------------+
|                        RELATION VS. FUNCTION MAPPING                        |
|                                                                             |
|       FUNCTION (Valid Mapping)               NOT A FUNCTION (Invalid)       |
|       ------------------------               ------------------------       |
|       Domain (x)     Range (y)               Domain (x)     Range (y)       |
|          ( 1 ) ------> ( 5 )                    ( 1 ) ------> ( 5 )         |
|          ( 2 ) ------> ( 8 )                    ( 2 ) --+---> ( 8 )         |
|          ( 3 ) --+                                      +---> ( 9 )         |
|                  +---> ( 10)                    ( 3 ) ------> ( 12)         |
|          ( 4 ) --+                                                          |
|                                                                             |
|   Each input maps to ONE output.            Input '2' maps to TWO outputs!  |
|   Inputs 3 & 4 share output 10 (OK).        Fails the function definition.  |
+-----------------------------------------------------------------------------+

The Vertical Line Test (VLT)

In the Cartesian coordinate plane, the geometric equivalent of the function definition is the Vertical Line Test:

[!IMPORTANT] The Vertical Line Test: A curve in the $xy$-plane represents the graph of $y$ as a function of $x$ if and only if no vertical line intersects the curve at more than one point.

• If even a single vertical line $x = c$ intersects the curve two or more times, the relation is not a function (because the single input $x = c$ yields multiple $y$-outputs).
• Circles ($x^2 + y^2 = r^2$), ellipses, and sideways parabolas ($x = y^2$) fail the Vertical Line Test.

---

2. Function Notation & Algebraic Evaluation

Function notation replaces the dependent variable $y$ with the symbol $f(x)$, read as "$f$ of $x$". The notation emphasizes that the output value depends upon the input value placed inside the parentheses.

Numerical Evaluation

To evaluate $f(c)$ for a constant $c$, substitute $c$ for every occurrence of the variable $x$ and simplify using the standard order of operations (PEMDAS).

Worked Example: Given $f(x) = 2x^2 - 5x + 3$, evaluate $f(-3)$.

$f(-3) = 2(-3)^2 - 5(-3) + 3 = 2(9) + 15 + 3 = 18 + 15 + 3 = 36$

Algebraic Expression Evaluation

When the input is an algebraic expression rather than a number, substitute the entire expression enclosed in parentheses into every variable position.

Worked Example: For $f(x) = 3x^2 - 4x + 1$, find and simplify $f(2t - 1)$.

f(2t - 1) &= 3(2t - 1)^2 - 4(2t - 1) + 1 \\ &= 3(4t^2 - 4t + 1) - 8t + 4 + 1 \\ &= 12t^2 - 12t + 3 - 8t + 5 \\ &= 12t^2 - 20t + 8 \end{aligned}$$ ### The Difference Quotient A fundamental algebraic construct frequently tested on the CLEP is the **Difference Quotient**, which represents the average rate of change of $f(x)$ over the interval $[x, x + h]$: $$\text{Difference Quotient} = \frac{f(x + h) - f(x)}{h} \quad (h \neq 0)$$ #### Step-by-Step Difference Quotient Evaluation Find the simplified difference quotient for $f(x) = 2x^2 - 3x + 4$: 1. **Step 1: Compute $f(x + h)$:** $$f(x + h) = 2(x + h)^2 - 3(x + h) + 4 = 2(x^2 + 2xh + h^2) - 3x - 3h + 4 = 2x^2 + 4xh + 2h^2 - 3x - 3h + 4$$ 2. **Step 2: Form the numerator $[f(x + h) - f(x)]$:** $$(2x^2 + 4xh + 2h^2 - 3x - 3h + 4) - (2x^2 - 3x + 4) = 4xh + 2h^2 - 3h$$ *(Notice that all terms not containing $h$ cancel out completely).* 3. **Step 3: Divide by $h$ and factor:** $$\frac{4xh + 2h^2 - 3h}{h} = \frac{h(4x + 2h - 3)}{h} = 4x + 2h - 3$$ --- ## 3. Algebraic Determination of Domain and Range Unless explicitly stated otherwise, the domain of a function $f$ is assumed to be the **natural domain**: the set of all real numbers $\mathbb{R} = (-\infty, \infty)$ for which the algebraic expression produces a real number output. ### The Three Major Domain Restrictions | Function Type | Algebraic Structure | Restriction Condition | Core Domain Rule | | :--- | :--- | :--- | :--- | | **Rational Functions** | $f(x) = \frac{P(x)}{Q(x)}$ | Denominator cannot equal zero | Set $Q(x) \neq 0$ and solve for excluded points | | **Even Radical Functions** | $f(x) = \sqrt[n]{g(x)}$ ($n$ even) | Radicand must be non-negative | Set $g(x) \ge 0$ and solve the inequality | | **Logarithmic Functions** | $f(x) = \log_b(g(x))$ | Argument must be strictly positive | Set $g(x) > 0$ and solve the strict inequality | > [!NOTE] > **Odd Roots Have No Radicand Restrictions:** > Cube roots and other odd roots ($\sqrt[3]{x}$, $\sqrt[5]{x}$) are fully defined for all real numbers, including negative values (e.g., $\sqrt[3]{-8} = -2$). Therefore, the domain of $f(x) = \sqrt[3]{2x - 7}$ is $(-\infty, \infty)$. ### Worked Example: Combined Domain Restrictions Find the domain of $f(x) = \frac{\sqrt{3x + 12}}{x^2 - 25}$ in interval notation. 1. **Numerator Radical Restriction:** The radicand under the square root must be non-negative: $$3x + 12 \ge 0 \implies 3x \ge -12 \implies x \ge -4$$ 2. **Denominator Non-Zero Restriction:** The denominator cannot equal zero: $$x^2 - 25 \neq 0 \implies (x - 5)(x + 5) \neq 0 \implies x \neq 5 \text{ and } x \neq -5$$ 3. **Synthesizing Restrictions:** - The radical condition requires $x \in [-4, \infty)$. - The point $x = -5$ is already outside $[-4, \infty)$, so it creates no additional cut inside the interval. - The point $x = 5$ lies within $[-4, \infty)$ and must be excluded. 4. **Final Domain:** **$[-4, 5) \cup (5, \infty)$**. --- ## 4. Survey of Function Families on the CLEP | Function Family | General Algebraic Form | Standard Natural Domain | Standard Range (Parent) | Defining Graphical Feature | | :--- | :--- | :--- | :--- | :--- | | **Linear** | $f(x) = mx + b$ ($m \neq 0$) | $(-\infty, \infty)$ | $(-\infty, \infty)$ | Straight line with constant slope $m$ | | **Quadratic (Polynomial)** | $f(x) = ax^2 + bx + c$ ($a > 0$) | $(-\infty, \infty)$ | $[k, \infty)$ where $k = f(-b/2a)$ | Parabola opening upward/downward | | **Square Root (Radical)** | $f(x) = \sqrt{x}$ | $[0, \infty)$ | $[0, \infty)$ | Half-parabola opening to the right | | **Rational (Reciprocal)** | $f(x) = \frac{1}{x}$ | $(-\infty, 0) \cup (0, \infty)$ | $(-\infty, 0) \cup (0, \infty)$ | Hyperbola with asymptotes at $x = 0, y = 0$ | | **Exponential** | $f(x) = b^x$ ($b > 0, b \neq 1$) | $(-\infty, \infty)$ | $(0, \infty)$ | Strictly positive curve; horizontal asymptote $y = 0$ | | **Logarithmic** | $f(x) = \log_b(x)$ | $(0, \infty)$ | $(-\infty, \infty)$ | Vertical asymptote at $x = 0$; inverse of exponential | | **Absolute Value** | $f(x) = |x|$ | $(-\infty, \infty)$ | $[0, \infty)$ | V-shaped graph with vertex at $(0, 0)$ | --- ## 5. Piecewise-Defined Functions A **piecewise-defined function** is a function governed by different algebraic formulas on distinct intervals of its domain. ### General Form and Evaluation Algorithm $$f(x) = \begin{cases} g(x) & \text{if } x < a \\ h(x) & \text{if } a \le x \le b \\ k(x) & \text{if } x > b \end{cases}$$ **Evaluation Procedure:** 1. Take the given input value $c$. 2. Locate the specific condition (sub-interval) satisfied by $c$. 3. Substitute $c$ **exclusively** into that single corresponding branch formula. 4. **Never** substitute an input into more than one branch. ### Worked Example: Piecewise Evaluation Consider the piecewise function $f(x)$ defined by: $$f(x) = \begin{cases} 2x + 5 & \text{if } x < -2 \\ 3 - x^2 & \text{if } -2 \le x \le 4 \\ \frac{12}{x} & \text{if } x > 4 \end{cases}$$ - **Evaluate $f(-4)$:** Because $-4 < -2$, use branch 1: $f(-4) = 2(-4) + 5 = -8 + 5 = -3$. - **Evaluate $f(-2)$:** Because $-2$ satisfies $-2 \le x \le 4$, use branch 2: $f(-2) = 3 - (-2)^2 = 3 - 4 = -1$. - **Evaluate $f(4)$:** Because $4$ satisfies $-2 \le x \le 4$, use branch 2: $f(4) = 3 - (4)^2 = 3 - 16 = -13$. - **Evaluate $f(6)$:** Because $6 > 4$, use branch 3: $f(6) = \frac{12}{6} = 2$. --- ## 6. Common CLEP Traps & Strategic Checkpoints - **Trap 1: Confusing $f(x + h)$ with $f(x) + h$:** $f(x + h)$ requires inserting $(x + h)$ into every variable slot. Writing $f(x) + h = (3x^2 - 4x + 1) + h$ instead of $3(x + h)^2 - 4(x + h) + 1$ is a fatal algebraic error. - **Trap 2: Restricting the Domain of Odd Roots:** Do not restrict $\sqrt[3]{x - 5} \ge 0$. Odd radicals accept negative radicands; their domain is all real numbers unless a denominator or logarithm is also present. - **Trap 3: Non-Strict vs. Strict Inequalities in Domain Setup:** - Even radical in numerator: $g(x) \ge 0$ (bracket $[$ allowed). - Even radical in denominator: $g(x) > 0$ (strict parenthesis $($, since denominator cannot equal 0). - Logarithm argument: $g(x) > 0$ (always strict parenthesis $($, since $\log_b(0)$ is undefined). - **Trap 4: Misreading Piecewise Boundary Endpoints:** Pay close attention to strict ($<, >$) versus inclusive ($\le, \ge$) inequality symbols at boundary transition points.