Algebra Functions Course Updated

Master functions with visual explainers, step‑by‑step worked examples, interactive tools, and quizzes. Learn how to find domain & range, analyze graphs, build composites, invert functions, and model real problems.

Try the Calculator Jump to Challenge Quiz

Understanding Functions

A function assigns exactly one output to each valid input. We write f(x) for the output when the input is x. Key ideas:

Domain: all allowed inputs.
Range: outputs produced by the rule.
Notation: y = f(x).
Graph: set of points (x, f(x)). Use the vertical line test to check function status.

Common families: linear, quadratic, exponential, logarithmic, rational, trigonometric, piecewise.

Tip: Always check for restrictions like division by zero or even roots of negatives.

inputs ↦ outputs one x → one y function rule

Learning Objectives

Determine domain & range (including logs/roots/rationals).
Describe transformations from a parent function.
Compute intercepts and average rate of change.
Build f∘g and test one‑to‑one with the HLT.
Find inverses (and when to restrict the domain).
Analyze asymptotes and end behavior.

Key Function Types — Explainers & Examples

1) Linear: f(x) = mx + b

Straight line with slope m and intercept b. Domain: all reals; Range: all reals.

Example: f(x)=3x−2 → f(4)=10. Zero at x=2/3. Increasing if m>0.

2) Quadratic: f(x) = ax² + bx + c

Parabola opening up if a>0, down if a<0. Vertex at x = −b/(2a). Axis of symmetry through the vertex.

Example: f(x)=x²−4x+3 = (x−1)(x−3). Zeros at x=1 and x=3. Minimum at x=2.

3) Exponential: f(x)=a·b^x

Constant percentage growth/decay. Horizontal asymptote y=0 if a≠0. Domain: all reals. Range: y has sign of a.

Example: P(t)=500·(1.08)^t grows 8% per time unit.

4) Logarithmic: f(x)=a·log_b(x)+c

Inverse of b^x. Domain: x>0. Vertical asymptote x=0. Useful for orders of magnitude (pH, decibels).

5) Rational: f(x)=P(x)/Q(x)

Undefined where Q(x)=0 → vertical asymptotes. Horizontal/slant asymptotes from polynomial degrees.

6) Piecewise

Different rules on different intervals. Carefully state interval boundaries and include closed/open dots in graphs.

Linear — Quick Solve

Find the x‑intercept of f(x)=3x−2.

Set f(x)=0 ⇒ 3x−2=0
Solve ⇒ x=2/3.

Rate of change is constant: slope m=3.

Quadratic — Vertex & Zeros

For x^2−4x+3:

Vertex at x=−b/(2a)=2 ⇒ (2,−1).
Factor ⇒ (x−1)(x−3) ⇒ zeros at 1 and 3.

Use completing the square to reach vertex form.

Exponential vs. Log

Solve 3^x=20.

Take logs ⇒ x=(ln 20)/(ln 3) ≈ 2.7268.
Check by substitution in calculator.

Exponential growth beats polynomial growth for large x.

Transformations Cheat‑Sheet

f(x)+k → up by k
f(x−h) → right by h
a·f(x) → vertical stretch by |a|
f(bx) → horizontal shrink by |b|
−f(x)/f(−x) → reflect

Tip: Apply horizontal changes (inside x) before vertical ones when sketching.

Quick Practice

Enter a value for x:

f(x) =

Your Progress

We save your correct answers locally on your device.

Solved: 0 | Streak: 0

Explore More Topics

Function Composition — Combine rules using (f ∘ g)(x).
Inverse Functions — Reverse inputs/outputs, f⁻¹(x).
Piecewise Functions — Multiple rules on intervals.
Domain & Range — Valid inputs and outputs.
Vertical Line Test — Is it a function?
Transformations — Shifts, reflections, stretches.

Deep‑Dive: Domain & Range Playbook

Denominators ≠ 0 (exclude roots of Q(x)).
Even roots need non‑negative radicands.
Logs need positive inputs.
Piecewise: enforce each interval condition.

Example: f(x)= √(x−2) / ( x(x−3) )

Requirements: x−2 ≥ 0 and x≠0,3 ⇒ domain [2,∞) without {0,3}.

Range tip: Solve y=f(x) for x, then check feasibility. Consider horizontal asymptotes and turning points.

For quadratics, the range begins at the vertex y‑value if a>0 (minimum) or extends downward if a<0.

Functions Calculator

Enter a function f(x): Enter a value for x: Enter g(x) for composition f(g(x)): Enter interval [a, b] for integration:

Result:

Graph:

Rational Functions

A rational function has the form f(x)=P(x)/Q(x) with polynomials P and Q (Q ≠ 0).

Domain: exclude roots of Q(x).
Vertical asymptotes: roots of Q(x) that don’t cancel.
Horizontal asymptote: compare degrees of P and Q.
Oblique (slant): deg(P)=deg(Q)+1 → divide P by Q.

Numerator P(x): Denominator Q(x):

Composite Functions

(f ∘ g)(x) = f(g(x)). Substitute g(x) into f(x).

Outer function f(x): Inner function g(x):

Even & Odd Functions

Even: f(-x)=f(x)

Odd: f(-x)=-f(x)

Graphs of sin(x) and cos(x)

sin(x) is odd: sin(−x)=−sin(x).

cos(x) is even: cos(−x)=cos(x).

Operations on Functions

Given functions f and g, define for all x in their common domain:

(f + g)(x) = f(x) + g(x)
(f − g)(x) = f(x) − g(x)
(f · g)(x) = f(x)·g(x)
(f / g)(x) = f(x) / g(x) where g(x) ≠ 0

f(x): g(x): x:

End Behavior, Limits & Asymptotes

End behavior describes what f(x) does as x approaches plus or minus infinity. For polynomials, the leading term dominates. For rational functions, compare degrees of numerator and denominator.

Horizontal asymptote: y = L if the limit of f(x) as x→±∞ equals L.
Vertical asymptote at x = a if the limit of f(x) as x→a from left or right is infinite.
Slant asymptote: divide P(x) by Q(x) when degree(P)=degree(Q)+1.

Rational f(x)=P/Q: /

Holes vs. vertical asymptotes: If a factor cancels between P and Q, you have a hole (removable discontinuity) at that x. If it doesn’t cancel, you have a vertical asymptote.

Example: ((x−2)(x+1))/((x−2)(x−4)) ⇒ hole at x=2, VA at x=4.

Degree rules: deg(P)<deg(Q) ⇒ HA y=0; deg(P)=deg(Q) ⇒ HA y=leadCoef(P)/leadCoef(Q); deg(P)=deg(Q)+1 ⇒ slant.

Horizontal Line Test (One-to-One)

A function is one-to-one if every horizontal line intersects its graph at most once. One-to-one functions have inverses on their (full) domains.

Try:

Continuity & Differentiability

Continuous functions can be drawn without lifting your pencil. Differentiable functions have a defined slope at each interior point of their domain.

Removable discontinuity: a hole (limit exists but f(a) missing or different).
Jump discontinuity: left/right limits are finite but unequal.
Infinite discontinuity: vertical asymptote.

Enter piecewise (JS style):

How to Find an Inverse Function

Write y=f(x).
Swap x and y.
Solve for y.
Rename y as f⁻¹(x). (Restrict the domain if f is not one‑to‑one.)

Example: f(x)=2x+4 ⇒ x=2y+4 ⇒ y=(x−4)/2 ⇒ f⁻¹(x)=(x−4)/2.

For f(x)=x^2, restrict to x≥0 (or x≤0) before inverting to get f⁻¹(x)=√x.

Common Function Forms

Family	Formula	Domain	Range	Key Features
Linear	mx+b	R	R	slope m, intercept b
Quadratic	ax^2+bx+c	R	depends on a	vertex, symmetry axis
Exponential	a·b^x	R	sign of a	horizontal asymptote y=0
Logarithmic	a·log_b(x)+c	x>0	R	vertical asymptote x=0
Rational	P(x)/Q(x)	x: Q(x)≠0	R except restricted values	vertical/horizontal/slant asymptotes
Absolute value	\|x\|	R	[0,∞)	corner at 0, even
Power	x^n	R (n integer)	depends on n	even/odd parity
Trig	sin x, cos x	R	[−1,1]	periodic

Parent functions help you sketch transformations quickly. Start from the parent, then apply shifts, stretches, and reflections.