Functions & Algebra: Domain, Graphing, and Key Techniques

Posted on Jan 28, 2026 in Mathematics

Function Basics

Definition: A function assigns exactly one output for each input.

Function notation: f(x)

Vertical Line Test: If any vertical line intersects a graph more than once, the relation is not a function.

Using Tables to Graph (by Hand)

Steps:

• Choose x-values (use symmetry if possible).
• Plug the x-values into the function.
• Create a table of (x, f(x)).
• Plot the points.
• Connect points smoothly to represent the graph.

Always check:

• x-intercepts
• y-intercept
• asymptotes (if any)

Domain Rules (Very Important)

• Polynomials: domain is all real numbers (−∞, ∞).
• Rational functions: denominator ≠ 0.
• Square roots: expression inside the radical ≥ 0 (for real-valued principal root).
• Logarithms: argument > 0.

Range

Determine the range by examining the graph.

Quadratic functions:

• If the parabola opens up, the range starts at the minimum value.
• If the parabola opens down, the range ends at the maximum value.
• Use the vertex to identify extrema for quadratics.

Intercepts

• x-intercepts: set y = 0 and solve for x.
• y-intercept: set x = 0 and evaluate y.

Increasing / Decreasing

Increasing: the graph rises from left to right.

Decreasing: the graph falls from left to right.

Quadratics:

• If the parabola opens up: decreasing → vertex → increasing.
• If the parabola opens down: increasing → vertex → decreasing.

Transformations (Memorize)

Given y = a·f(x − h) + k:

• h: horizontal shift (opposite direction of the sign; e.g., x − h shifts right by h).
• k: vertical shift (up for +k, down for −k).
• a: vertical stretch/shrink (|a| > 1 stretches, 0 < |a| < 1 shrinks).
• Negative a: reflection over the x-axis.

Quadratic Functions

Standard form: y = ax^2 + bx + c

Direction:

• Opens up if a > 0.
• Opens down if a < 0.

Vertex (axis of symmetry): x = −b / (2a).

Vertex form: y = a(x − h)^2 + k (vertex = (h, k)).

Solving Quadratics

Factoring

• Set the equation = 0.
• Factor the polynomial.
• Set each factor = 0 and solve.

Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

Discriminant (Δ = b² − 4ac):

• Δ > 0: two distinct real solutions.
• Δ = 0: one real (repeated) solution.
• Δ < 0: two complex solutions (no real solutions).

Polynomial Functions

Degree: highest exponent.

End behavior:

• Even degree, positive leading coefficient → both ends up.
• Even degree, negative leading coefficient → both ends down.
• Odd degree, positive leading coefficient → left down, right up.
• Odd degree, negative leading coefficient → left up, right down.

Zeros / roots: x-intercepts. Multiplicity:

• Odd multiplicity → graph crosses the x-axis at the root.
• Even multiplicity → graph touches and bounces at the root.

Rational Functions

Form: f(x) = p(x) / q(x)

• Domain: values where the denominator ≠ 0.
• Vertical asymptote: denominator = 0 (factor does not cancel).
• Hole: occurs where a factor cancels between numerator and denominator.
• Horizontal asymptote:
  • If degree(numerator) < degree(denominator): y = 0.
  • If degrees are equal: y = ratio of leading coefficients.
  • If degree(numerator) > degree(denominator): there may be a slant (oblique) asymptote (if applicable).

Logarithmic Functions

Example: y = log(x)

• Domain: x > 0.
• Vertical asymptote: x = 0.
• Logarithms are the inverse of exponentials.

Log rules (if included):

• log(ab) = log a + log b
• log(a/b) = log a − log b
• log(a^n) = n · log a

Solving Inequalities

Graphing Method

• Move all terms to one side.
• Factor when possible.
• Find zeros (critical points).
• Create a sign chart (test intervals).
• Write the solution in interval notation.

Sign Chart Tips

• Test points between zeros to determine sign.
• When multiplying or dividing by a negative number, flip the inequality sign.

Interval Notation