3.5 Function Notation, Evaluating Functions, and Graph Analysis

3.5 Function Notation, Evaluating Functions, and Graph Analysis

Quick Answer: A function is a rule that assigns each element of an input set (the domain) to exactly one element of an output set (the range). Function notation f(x) replaces y to explicitly represent the rule applied to the independent variable x. Linear functions exhibit a constant slope m = Δy/Δx, with parallel lines having identical slopes and perpendicular lines having negative reciprocal slopes. Analyzing function graphs involves locating zeros (x-intercepts), initial values (y-intercepts), extrema (maxima and minima), and determining intervals over which the function increases or decreases.

Formal Definition of a Function, Domain, and Range

A relation is any set of ordered pairs (x, y). A function is a specialized relation with a strict uniqueness requirement:

Function Requirement: For every element x in the domain, there exists exactly one corresponding element y in the range. No single input value may produce multiple different output values.

Identifying Functions Across Representations:

• Set of Ordered Pairs: Inspect the first coordinates for repeating values. The set {(1, 4), (2, 7), (3, 4)} is a valid function (the distinct inputs 1 and 3 can share the output 4). The set {(1, 4), (1, 9), (2, 5)} is not a function because input 1 maps to two different outputs (4 and 9).
• Mapping Diagrams: A relation is a function if every element in the domain bubble has exactly one arrow pointing away from it.
• Graphical Test (The Vertical Line Test): If any vertical line can be drawn across the Cartesian plane that intersects a graph in more than one point, the graph does not represent a function of x. A circle or vertical line fails the test; a non-vertical line or standard parabola passes.

Domain Restrictions in the Real Number System

When determining the domain of an algebraic function y = f(x) over ℝ, assume all real numbers are permissible except those violating two fundamental mathematical constraints:

1. Division by Zero: Denominators cannot equal zero. Set every denominator equal to zero and exclude those values from the domain.
2. Even Roots of Negative Numbers: The expression under any even radical (such as a square root √) must be greater than or equal to zero. Solve radicand ≥ 0.

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Function Notation and Operations

In function notation, f(x) is read 'f of x'. The letter f names the function rule, and the symbol in parentheses represents the input variable. It does not indicate multiplication of f by x. It is a unified symbol representing the output value y.

Evaluating Functions with Expressions

Evaluating a function involves substituting the input expression into every occurrence of the independent variable:

• If f(x) = 3x² - 5x + 2: f(-2) = 3(-2)² - 5(-2) + 2 = 3(4) + 10 + 2 = 24.
• Evaluating with a binomial input: f(x + h) = 3(x + h)² - 5(x + h) + 2 = 3(x² + 2xh + h²) - 5x - 5h + 2 = 3x² + 6xh + 3h² - 5x - 5h + 2.

Composite Functions: (f ∘ g)(x) = f(g(x))

In a composite function, the output of the inner function g(x) serves as the direct input to the outer function f:

1. To evaluate f(g(k)) for a numerical value k, evaluate the inner value g(k) first, then substitute that numerical result into f.
2. To find the algebraic rule f(g(x)), substitute the entire expression g(x) into every x in f(x).

Worked Example: Composite Function Evaluation Let f(x) = 2x² - 1 and g(x) = 3x + 4. Find f(g(-2)) and g(f(x)).

Step 1: Evaluate f(g(-2)). First evaluate inner function g(-2): g(-2) = 3(-2) + 4 = -6 + 4 = -2. Now evaluate outer function f with the result -2: f(-2) = 2(-2)² - 1 = 2(4) - 1 = 7. Thus, f(g(-2)) = 7.

Step 2: Find algebraic rule g(f(x)). g(f(x)) = g(2x² - 1) = 3(2x² - 1) + 4 = 6x² - 3 + 4 = 6x² + 1.

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Linear Functions and Coordinate Geometry

A linear function is characterized by a constant rate of change (constant slope).

The Slope Formula

Given two distinct points (x₁, y₁) and (x₂, y₂) on a non-vertical line: m = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx = (rise) / (run)

• Positive slope (m > 0): Line rises from left to right.
• Negative slope (m < 0): Line falls from left to right.
• Zero slope (m = 0): Horizontal line (equation: y = c).
• Undefined slope: Vertical line (equation: x = c); not a function.

Forms of Linear Equations

1. Slope-Intercept Form: y = mx + b
   • m is the slope, and (0, b) is the y-intercept.
2. Point-Slope Form: y - y₁ = m(x - x₁)
   • Ideal when given the slope and any arbitrary coordinate point (x₁, y₁).
3. Standard Form: Ax + By = C (where A, B, C are integers, and A ≥ 0)
   • Slope is -A/B; y-intercept is C/B; x-intercept is C/A.

Parallel and Perpendicular Lines

The geometric orientation between two lines is governed entirely by their slopes:

• Parallel Lines: Two lines are parallel if and only if they possess identical slopes and different y-intercepts: m₁ = m₂ and b₁ ≠ b₂.
• Perpendicular Lines: Two lines intersect at a 90° right angle if and only if their slopes are negative reciprocals of each other: m₁ · m₂ = -1 or m₂ = -1 / m₁ (for non-vertical, non-horizontal lines).
  • If m₁ = 3/4, then m₂ = -4/3.
  • If m₁ = -5, then m₂ = +1/5.

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Graphical Analysis and Landmark Features

Interpreting function graphs requires identifying critical structural landmarks:

Vertex of a Quadratic Function

For a quadratic function in standard form f(x) = ax² + bx + c:

• The x-coordinate of the vertex is x_v = -b / (2a).
• The y-coordinate is obtained by evaluating the function at that point: y_v = f(-b / (2a)).
• If a > 0, the minimum value of the function is y_v.
• If a < 0, the maximum value of the function is y_v.

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TSIA2 Exam Traps & Strategic Checkpoints

• Trap 1: The Notation Distributive Myth. Treating f(a + b) as f · a + f · b. Function notation denotes an operation, not a multiplier. For example, if f(x) = x², f(2 + 3) = f(5) = 25, whereas f(2) + f(3) = 4 + 9 = 13.
• Trap 2: Reversing the Coordinates in Slope. Writing (x₂ - x₁) / (y₂ - y₁) instead of (y₂ - y₁) / (x₂ - x₁). Vertical change (y) is always in the numerator: 'rise over run'.
• Trap 3: Incomplete Perpendicular Slopes. Changing only the sign without inverting (e.g., claiming perpendicular to 3 is -3), or inverting without changing the sign (claiming perpendicular to 3 is 1/3). Perpendicular slopes require both opposite sign and reciprocal inversion (-1/3).
• Trap 4: Stating Intervals Using y-Values. Intervals of increase and decrease are always expressed in terms of the x-values over which the behavior occurs, never the y-values.