Comprehensive Guide to Functions, Parent Function Families, and Function Evaluation

Foundations of Functions and the Vertical Line Test

• Mathematical relationships and functions serve as the primary framework for modeling real-world phenomena, enabling the recognition of patterns, formulation of predictions, and analysis of graphs.
• Definition of a Function:
  • A function is a relation that assigns exactly one output for each input.
  • In algebraic terms, for every value of $x$, there exists only one corresponding value of $y$.
• Function Notation:
  • A function is commonly designated using the notation $f(x)$, which is verbally articulated as "$f$ of $x$".
• The Vertical Line Test:
  • A visual and geometric method utilized to verify whether a given relation graphed on a Cartesian plane qualifies as a function.
  • Rule: If a vertical line crosses the graph more than once at any point, the relation is not a function.
  • Conversely, if every possible vertical line intersects the graph at no more than one single point, the relation is a function.

• Analysis of Practice Relations (Evaluating for Function Status):
  • Relation 1 (Coordinate Graph):
  • Displays a continuous S-shaped monotonic curve passing through the origin.
  • Any vertical line drawn across the domain intersects the curve at exactly one point.
  • Classification: Function.
  • Relation 2 (Coordinate Graph):
  • Displays a circle centered at the origin $(0, 0)$ with a radius of $2$ units (equation $x^2 + y^2 = 4$).
  • A vertical line, such as the y-axis ($x = 0$), intersects the circle at two distinct points: $(0, 2)$ and $(0, -2)$.
  • Classification: Not A Function.
  • Relation 3 (Input-Output Table):
  • Data entries:
    • Input $x = 1$ corresponds to output $g(1) = 2$
    • Input $x = 2$ corresponds to output $g(2) = 4$
    • Input $x = 3$ corresponds to output $g(3) = 6$
    • Input $x = 4$ corresponds to output $g(4) = 8$
  • Every distinct input value of $x$ is paired with exactly one unique output value of $g(x)$.
  • Classification: Function.
  • Relation 4 (Coordinate Graph):
  • Displays a horizontal parabola opening to the right with its vertex on the y-axis.
  • Any vertical line drawn to the right of the vertex crosses both the upper and lower branches simultaneously at two points.
  • Classification: Not A Function.
  • Relation 5 (Input-Output Table):
  • Data entries:
    • Input $x = 5$ corresponds to output $h(5) = -4$
    • Input $x = 6$ corresponds to output $h(6) = -2$
    • Input $x = 7$ corresponds to output $h(7) = 0$
    • Input $x = 5$ corresponds to output $h(5) = 2$
  • The single input value $x = 5$ is mapped to two distinct output values: $-4$ and $2$.
  • Classification: Not A Function.
  • Relation 6 (Coordinate Graph):
  • Displays a smooth, oscillating continuous wave across the coordinate plane.
  • Every vertical line drawn across the graph intersects the curve at exactly one point.
  • Classification: Function.

Parent Functions and Their Essential Characteristics

• Definition and Purpose:
  • A parent function represents the simplest, most fundamental form of a function within a specific function family.
  • It demonstrates the general shape, end behavior, and key geometric features of that family in the absence of any transformations (such as dilations, reflections, or translations).
  • Mastery of parent functions provides a reference baseline to understand, analyze, and predict how equations and graphs alter when parameters are modified.

• Overview of the Twelve Fundamental Parent Functions:
  • Linear Function:
  • Equation: $f(x) = x$
  • Characteristics: A straight diagonal line passing through the origin $(0, 0)$ with a constant slope of $1$, spanning Quadrant III to Quadrant I.
  • Quadratic Function:
  • Equation: $f(x) = x^2$
  • Characteristics: A symmetric U-shaped parabola with a minimum vertex located at the origin $(0, 0)$, opening upward into Quadrant II and Quadrant I.
  • Cubic Function:
  • Equation: $f(x) = x^3$
  • Characteristics: An origin-symmetric S-shaped curve passing through $(0, 0)$ with an inflection point at the origin, extending downwards into Quadrant III and upwards into Quadrant I.
  • Absolute Value Function:
  • Equation: $f(x) = |x|$
  • Characteristics: A symmetric V-shaped graph consisting of two linear rays meeting at a sharp vertex at the origin $(0, 0)$, opening upward into Quadrant II and Quadrant I.
  • Square Root Function:
  • Equation: $f(x) = \sqrt{x}$
  • Characteristics: A half-parabola oriented horizontally, originating at the endpoint $(0, 0)$ and curving continuously through Quadrant I; restricted to non-negative domain values ($x \ge 0$).
  • Cube Root Function:
  • Equation: $f(x) = \sqrt[3]{x}$
  • Characteristics: An extended sideways S-shaped curve passing through the origin $(0, 0)$, with unrestricted real domain spanning Quadrants III and I.
  • Reciprocal Function:
  • Equation: $f(x) = \frac{1}{x}$
  • Characteristics: A hyperbola consisting of two symmetric, disjoint curves located in Quadrants I and III, asymptotically approaching the axes at $x = 0$ and $y = 0$.
  • Exponential Function:
  • Equation: $f(x) = e^x$
  • Characteristics: A curve demonstrating rapid exponential growth to the right, crossing the y-axis at $(0, 1)$, with a horizontal asymptote along the negative x-axis ($y = 0$).
  • Logarithmic Function:
  • Equation: $f(x) = \log(x)$
  • Characteristics: The inverse curve to exponential functions, crossing the x-axis at $(1, 0)$, curving gradually upward in Quadrant I, with a vertical asymptote along the negative y-axis ($x = 0$).
  • Sinusoidal Function:
  • Equation: $f(x) = \sin(x)$
  • Characteristics: A smooth, continuous periodic wave oscillating symmetrically between $-1$ and $1$, passing through the origin $(0, 0)$.
  • Constant Function:
  • Equation: $f(x) = c$
  • Characteristics: A perfectly horizontal straight line with a constant slope of $0$, intersecting the vertical axis at $(0, c)$, yielding identical outputs for all inputs.
  • Step Function (Greatest Integer / Floor Function):
  • Equation: $f(x) = [x]$
  • Characteristics: A discontinuous stair-step graph composed of discrete horizontal segments, each featuring a solid closed endpoint on the left and an open circular endpoint on the right.

Evaluating Functions for Numeric Values and Applied Contexts

• Core Definition:
  • Evaluating a function involves calculating the specific output value for a designated input by directly substituting a real number in place of the independent variable $x$.

• Algebraic Evaluation with Numeric Inputs:
  • Rational and Radical Combination (Problem 7):
  • Given Function: $f(x) = \frac{1}{3}x^2 - 2\sqrt{x}$
  • Target: Evaluate $f(9)$
  • Step 1: Substitute $9$ for $x$:       $f(9) = \frac{1}{3}(9)^2 - 2\sqrt{9}$
  • Step 2: Simplify the exponent and square root:       $(9)^2 = 81$$\sqrt{9} = 3$
  • Step 3: Perform scalar multiplications:       $\frac{1}{3}(81) = 27$$2(3) = 6$
  • Step 4: Subtract the terms:       $27 - 6 = 21$
  • Result:       $f(9) = 21$
  • Fractional Rational and Radical Expression (Problem 8):
  • Given Function: $h(x) = \frac{\sqrt{2x+3}}{6x-5}$
  • Target: Evaluate $h(\frac{1}{2})$
  • Step 1: Substitute $\frac{1}{2}$ into the numerator and denominator:       $h(\frac{1}{2}) = \frac{\sqrt{2(\frac{1}{2})+3}}{6(\frac{1}{2})-5}$
  • Step 2: Simplify the numerator expression:       $2(\frac{1}{2}) + 3 = 1 + 3 = 4$$\sqrt{4} = 2$
  • Step 3: Simplify the denominator expression:       $6(\frac{1}{2}) - 5 = 3 - 5 = -2$
  • Step 4: Divide the simplified components:       $\frac{2}{-2} = -1$
  • Result:       $h(\frac{1}{2}) = -1$
• Graphical Function Evaluation:
  • Coordinate Graph Reading (Problem 9):
  • Target: Find the value of $g(-4)$ from the graph of $g(x)$.
  • Procedure: Locate $-4$ along the horizontal x-axis and trace vertically upward to meet the curve.
  • The curve attains a vertical coordinate of $6$ when the input is $-4$.
  • Result:       $g(-4) = 6$
• Real-World Mathematical Modeling:
  • Quadratic Model: United States Population (Problem 10):
  • Mathematical Formula: $P(t) = 0.008t^2 + 1.115t + 78.4$
  • Variable Definitions:
    • $P$ = Total United States population measured in millions of people.
    • $t$ = Time elapsed in years since the baseline year 1900.
  • Objective: Estimate the population in the year 2030 to the nearest tenth of a million.
  • Step 1: Determine the elapsed time $t$:       $t = 2030 - 1900 = 130$
  • Step 2: Substitute $t = 130$ into the model:       $P(130) = 0.008(130)^2 + 1.115(130) + 78.4$
  • Step 3: Calculate powers and products:       $(130)^2 = 16900$$0.008(16900) = 135.2$$1.115(130) = 144.95$
  • Step 4: Sum all evaluated components:       $P(130) = 135.2 + 144.95 + 78.4 = 358.55$
  • Step 5: Round to the nearest tenth of a million:       $358.55 \approx 358.6$
  • Result: The estimated population in 2030 is $358.6\,\text{million}$.
  • Exponential Model: Vehicle Depreciation (Problem 11):
  • Mathematical Formula: $v(t) = 42000(0.82)^t$
  • Variable Definitions:
    • $v(t)$ = Monetary value of the vehicle in dollars.
    • $t$ = Time elapsed in years since the vehicle was purchased.
    • Base value of $0.82$ denotes an annual retention of $82\%$ of value ($18\%$ annual depreciation).
  • Objective: Calculate the value of $v(4)$ and explain its contextual meaning.
  • Step 1: Substitute $t = 4$ into the equation:       $v(4) = 42000(0.82)^4$
  • Step 2: Compute the exponential term:       $(0.82)^4 = 0.45212176$
  • Step 3: Multiply by the initial purchase price:       $42000 \times 0.45212176 = 18989.11392$
  • Step 4: Round to standard monetary format (two decimal places):       $v(4) = 18989.11$
  • Contextual Interpretation: After $4$ years, the car is worth $\$18989.11$.

Evaluating Functions with Algebraic Expressions and Transformation Foundations

• Conceptual Principles:
  • Beyond substituting single constants (e.g., $f(3)$), entire variable expressions—such as $f(2x)$ or $f(x + 1)$—can be inserted into a function's argument.
  • This procedure demonstrates how a function's output behaves when its input undergoes deliberate systematic shifts or scalings.
  • Algebraic substitutions form the direct structural foundation for geometric transformations, including horizontal stretching, compression, shifts, and axis reflections.
  • Fundamental Procedure: Replace every occurrence of the input variable with the substituted expression encapsulated within parentheses, expand powers algebraically, distribute multipliers and negative signs, and combine all like terms.

• Step-by-Step Expression Evaluations:
  • Scaling Input in a Quadratic Function (Problem 12):
  • Given Function: $f(x) = x^2 - 4x$
  • Target: Find and simplify $f(2x)$
  • Step 1: Substitute $(2x)$ into every variable instance:       $f(2x) = (2x)^2 - 4(2x)$
  • Step 2: Square the monomial term:       $(2x)^2 = 4x^2$
  • Step 3: Multiply the linear coefficient:       $-4(2x) = -8x$
  • Result:       $f(2x) = 4x^2 - 8x$
  • Horizontal Shift in a Reflected Quadratic (Problem 13):
  • Given Function: $g(x) = -x^2 + 2$
  • Target: Find and simplify $g(x + 1)$
  • Step 1: Substitute $(x + 1)$ for $x$:       $g(x + 1) = -(x + 1)^2 + 2$
  • Step 2: Expand the squared binomial:       $(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1$
  • Step 3: Enclose the trinomial in brackets and distribute the leading negative sign:       $-(x^2 + 2x + 1) = -x^2 - 2x - 1$
  • Step 4: Combine constant terms with $+2$:       $-x^2 - 2x - 1 + 2 = -x^2 - 2x + 1$
  • Result:       $g(x + 1) = -x^2 - 2x + 1$
  • Input Negation in a Radical Function (Problem 14):
  • Given Function: $h(x) = \sqrt{2x^2 - 9x}$
  • Target: Find and simplify $h(-x)$
  • Step 1: Substitute $(-x)$ into each position of $x$:       $h(-x) = \sqrt{2(-x)^2 - 9(-x)}$
  • Step 2: Evaluate the squared term ($(-x)^2 = x^2$):       $2(-x)^2 = 2x^2$
  • Step 3: Evaluate the linear term ($-9(-x) = +9x$):       $-9(-x) = 9x$
  • Step 4: Write the final simplified radicand:       $h(-x) = \sqrt{2x^2 + 9x}$
  • Result:       $h(-x) = \sqrt{2x^2 + 9x}$
  • Negative Scaling in a Higher-Order Polynomial (Problem 15):
  • Given Function: $k(x) = x^3 - 2x^2$
  • Target: Find and simplify $k(-3x)$
  • Step 1: Substitute $(-3x)$ into each instance of $x$:       $k(-3x) = (-3x)^3 - 2(-3x)^2$
  • Step 2: Evaluate the cubic power:       $(-3x)^3 = (-3)^3 x^3 = -27x^3$
  • Step 3: Evaluate the square and multiply by $-2$:       $(-3x)^2 = 9x^2$$-2(9x^2) = -18x^2$
  • Step 4: Combine the simplified terms:       $k(-3x) = -27x^3 - 18x^2$
  • Result:       $k(-3x) = -27x^3 - 18x^2$