Functions Calculator
Explore function domain, range, intercepts, extreme points and asymptotes step-by-step. Enter a function to analyze its properties and see its graph.
Function Analysis Results
| Function | f(x) = x² + 2x - 3 |
|---|---|
| Domain | All real numbers |
| Range | [-4, ∞) |
| X-Intercepts | (-3, 0), (1, 0) |
| Y-Intercept | (0, -3) |
| Extreme Points | Minimum at (-1, -4) |
| Asymptotes | None |
For polynomial functions like f(x) = x² + 2x - 3, the domain is all real numbers.
This is a quadratic function opening upwards. The vertex is at x = -b/(2a) = -2/(2*1) = -1.
Evaluating at x = -1: f(-1) = (-1)² + 2(-1) - 3 = 1 - 2 - 3 = -4
X-Intercepts: Set f(x) = 0 and solve x² + 2x - 3 = 0
Factoring: (x + 3)(x - 1) = 0 → x = -3 or x = 1
Y-Intercept: Evaluate f(0) = 0² + 2(0) - 3 = -3
The vertex of the parabola is the minimum point.
From earlier, vertex is at x = -1, f(-1) = -4
About Functions Calculator
The Functions Calculator is a powerful tool that allows you to analyze mathematical functions and visualize their behavior. It provides detailed information about a function's properties including its domain, range, intercepts, extreme points, and asymptotes.
How the Calculator Works
The calculator uses symbolic computation to analyze the function you provide:
- Parsing: The function expression is parsed into a computable format
- Domain: Determines all possible input values (x-values) for the function
- Range: Calculates all possible output values (y-values) the function can produce
- Intercepts: Finds where the function crosses the x-axis (roots) and y-axis
- Extreme Points: Identifies local maxima and minima using calculus techniques
- Asymptotes: Detects vertical, horizontal, and oblique asymptotes for rational functions
Supported Functions
The calculator can handle a wide variety of mathematical functions:
- Polynomials: x² + 2x - 3, 3x³ - 5x + 1
- Rational Functions: (x² - 1)/(x - 1), 1/(x² + 1)
- Exponential Functions: e^x, 2^(x + 1)
- Logarithmic Functions: ln(x), log₂(x + 3)
- Trigonometric Functions: sin(x), cos(2x + π/4)
- Absolute Value: |x - 2| + 1
- Square Roots: √(4 - x²)
Understanding Function Properties
Key concepts in function analysis:
- Domain: The set of all possible input values (x-values) for which the function is defined
- Range: The set of all possible output values (y-values) the function can produce
- Intercepts: Points where the graph crosses the x-axis (x-intercepts) or y-axis (y-intercept)
- Extreme Points: Local maxima (highest points) and minima (lowest points) on the graph
- Asymptotes: Lines that the graph approaches but never touches (for rational and some other functions)
Practical Applications
Function analysis is fundamental in many fields:
- Physics: Modeling motion, forces, and other physical phenomena
- Engineering: Designing systems and analyzing their behavior
- Economics: Modeling cost, revenue, and profit functions
- Biology: Population growth models and enzyme kinetics
- Computer Science: Algorithm analysis and complexity