13.3 Relations and Functions: Domain, Range, and Non-Linear Models

13.3 Relations and Functions: Domain, Range, and Non-Linear Models

Competencies 3.5 and 3.7 on the FTCE General Knowledge Mathematics subtest evaluate a candidate's understanding of mathematical relations, functional mappings, domain and range constraints, and fundamental non-linear models. In educational mathematics, functions serve as the universal language for modeling deterministic systems where inputs produce predictable outputs. Candidates must be able to classify relations, evaluate functions expressed in standard notation, define domain boundaries, and analyze the geometric and algebraic behaviors of quadratic and exponential functions.

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Relations, Functions, and the Uniqueness Criterion

A relation is formally defined as any set of ordered pairs $(x, y)$ that establishes a correspondence between an input set and an output set. A function is a specialized subset of relations governed by a strict uniqueness rule:

Definition of a Function: A relation is a function if and only if each element in the domain (input $x$) corresponds to exactly one element in the range (output $y$).

Diagnostic Tests for Functionality

To determine whether a relation represents a function across different formats, apply these standards:

• Set of Ordered Pairs / Tables: Examine the $x$-coordinates. If any $x$-value repeats with a different $y$-value, the relation is not a function. If all $x$-values are distinct, it is guaranteed to be a function. Note that repeating $y$-values do not violate the definition of a function (e.g., both $(-2, 4)$ and $(2, 4)$ can belong to the squaring function $f(x) = x^2$).
• Mapping Diagrams: Inspect the arrows departing from the input bubble (domain). If any single input value emits more than one arrow toward the output bubble, the relation fails the definition of a function.
• Coordinate Graphs — The Vertical Line Test (VLT): A visual curve or scatterplot represents a function of $x$ if and only if no vertical line can intersect the graph at more than one point. If even a single vertical line $x = c$ intersects the curve at two or more points, that single input $c$ maps to multiple outputs, disqualifying the relation. Circles, vertical parabolas opening sideways ($x = y^2$), and vertical line segments all fail the VLT.

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Domain and Range Analysis

The domain and range constitute the fundamental boundaries of any mathematical function:

• Domain: The set of all permissible input values ($x$-values or independent variables) for which the function produces a real number output.
• Range: The set of all resulting output values ($y$-values or dependent variables) generated by evaluating the function across its entire domain.

Primary Algebraic Domain Restrictions

In the real number system ($\mathbb{R}$), two primary operations impose domain restrictions:

1. Division by Zero is Undefined: Any input value that causes a denominator to equal zero must be strictly excluded from the domain.
   • Example: For $f(x) = \frac{5x + 2}{3x - 12}$, set $3x - 12 = 0 \implies 3x = 12 \implies x = 4$.
   • Domain: All real numbers except $x = 4$, written $\{x \in \mathbb{R} \mid x \neq 4\}$ or $(-\infty, 4) \cup (4, \infty)$.
2. Even Roots of Negative Numbers are Non-Real: An even radical (such as a square root $\sqrt{\phantom{x}}$) requires its radicand (the expression under the radical) to be greater than or equal to zero.
   • Example: For $g(x) = \sqrt{4x - 20}$, set $4x - 20 \ge 0 \implies 4x \ge 20 \implies x \ge 5$.
   • Domain: $\{x \in \mathbb{R} \mid x \ge 5\}$ or $[5, \infty)$.

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Function Notation and Algebraic Evaluation

Function notation replaces the dependent variable $y$ with the symbol $f(x)$, read as "$f$ of $x$." It emphasizes that the output depends directly on the input supplied within the parentheses.

• Evaluating $f(k)$ means replacing every occurrence of the independent variable $x$ in the formula with the expression or value $k$.

Worked Example: Multi-Step Function Evaluation

Consider the quadratic function $f(x) = 3x^2 - 4x + 7$.

• Part A: Evaluate for a negative numerical input, $f(-3)$. Substitute $-3$ for every $x$, maintaining parentheses:

  $$f(-3) = 3(-3)^2 - 4(-3) + 7 = 3(9) + 12 + 7 = 27 + 12 + 7 = 46$$

• Part B: Evaluate for an algebraic binomial input, $f(x + 2)$. Replace $x$ with $(x + 2)$:

  $$f(x + 2) = 3(x + 2)^2 - 4(x + 2) + 7$$

  Expand $(x + 2)^2 = x^2 + 4x + 4$:

  $$f(x + 2) = 3(x^2 + 4x + 4) - 4x - 8 + 7 = 3x^2 + 12x + 12 - 4x - 1 = 3x^2 + 8x + 11$$

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Non-Linear Models: Quadratic Functions

When the rate of change is not constant, linear models no longer suffice. The two most common non-linear models assessed on the FTCE are quadratic and exponential functions.

A quadratic function is a polynomial of degree 2, written in standard form as:

$$y = ax^2 + bx + c \quad (a \neq 0)$$

Its graph is a smooth, symmetric U-shaped curve known as a parabola.

Key Structural Features of Parabolas

• Direction of Opening:
  • If $a > 0$, the parabola opens upward ($\cup$). The vertex represents the absolute minimum point of the function.
  • If $a < 0$, the parabola opens downward ($\cap$). The vertex represents the absolute maximum point of the function.
• Axis of Symmetry: The vertical line that divides the parabola into mirror-image halves, given by: $$x = -\frac{b}{2a}$$
• Vertex: The peak or trough of the parabola, located at coordinate: $$(h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$
• $y$-Intercept: Setting $x = 0$ yields $y = c$, giving the coordinate $(0, c)$.

Worked Example: Real-World Projectile Trajectory

In a physics laboratory, a projectile is launched from an elevated platform. Its altitude $h(t)$ in feet after $t$ seconds is modeled by:

$$h(t) = -16t^2 + 64t + 80$$

1. Find the time to reach maximum height: Identify coefficients: $a = -16$, $b = 64$, $c = 80$. Because $a = -16 < 0$, the parabola opens downward and the vertex represents the maximum height. $$t_{\text{vertex}} = -\frac{b}{2a} = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2\text{ seconds}$$
2. Calculate the maximum height: Evaluate $h(2)$: $$h(2) = -16(2)^2 + 64(2) + 80 = -16(4) + 128 + 80 = -64 + 128 + 80 = 144\text{ feet}$$
3. Determine when the projectile strikes the ground: Set $h(t) = 0$ and solve for $t$: $$-16t^2 + 64t + 80 = 0$$ Divide the entire equation by $-16$: $$t^2 - 4t - 5 = 0 \implies (t - 5)(t + 1) = 0$$ Solutions are $t = 5$ or $t = -1$. Because time cannot be negative in physical projectile context, the projectile strikes the ground at $t = 5$ seconds.

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Non-Linear Models: Exponential Growth and Decay

An exponential function models processes where a quantity changes by a constant multiplicative factor over equal intervals of time:

$$y = a(b)^x \quad (a \neq 0, b > 0, b \neq 1)$$

• Initial Value ($a$): The baseline value when $x = 0$, representing the $y$-intercept $(0, a)$.
• Base / Growth or Decay Factor ($b$): The constant multiplier between consecutive integer values of $x$.

Growth vs. Decay Criteria

Linear vs. Exponential Growth Distinction

• Linear Growth: Adds a constant numerical quantity $\Delta y = m$ for every unit change in $x$ (constant additive rate).
• Exponential Growth: Multiplies by a constant ratio $\frac{y_{x+1}}{y_x} = b$ for every unit change in $x$ (constant percentage rate).

Worked Example: Equipment Depreciation (Exponential Decay)

A technical education program purchases a specialized laser cutter for $32,000. The machinery depreciates exponentially at an annual rate of $15\%$.

• Step 1: Identify model parameters. Initial value $a = 32,000$. Annual depreciation rate $r = 0.15$. Decay factor $b = 1 - r = 1 - 0.15 = 0.85$.
• Step 2: Construct the exponential model. $$V(t) = 32,000(0.85)^t$$
• Step 3: Calculate the asset value after 3 years. $$V(3) = 32,000(0.85)^3 = 32,000(0.614125) = 19,652$$

After 3 years, the equipment retains a market value of $19,652.

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Strategic Summary of Mathematical Models

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Frequent FTCE Exam Traps and Best Practices

• Input Duplication in Functions: Always check whether a single $x$ has multiple $y$ values. Multiple $x$ values sharing the same $y$ value is completely valid for functions.
• Sign of Vertex Axis of Symmetry: In $x = -\frac{b}{2a}$, remember the leading negative sign. If $b$ is negative, $-\frac{-b}{2a}$ becomes positive.
• Exponential Base Inversion: An annual decrease of $18\%$ means the decay base is $b = 1 - 0.18 = 0.82$, not $0.18$. Writing $y = a(0.18)^x$ would model an immediate loss of $82\%$ rather than $18\%$.