Composite Function Calculator
Calculate composite functions f(g(x)) and g(f(x)) with step-by-step solutions. Our calculus calculator analyzes function composition, domains, and provides detailed mathematical explanations.
Last updated: October 19, 2025
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Use x as variable. Supported: x^2, sin(x), cos(x), ln(x), sqrt(x), +, -, *, /
Second function for composition
Point at which to evaluate the composite function
Composition Results
Composite Function:
(x^2 + 1) + 1
Value at x = 2:
6.0000
Domain Considerations:
Real numbers (restrictions may apply)
Analysis:
The composite function f(g(x)) is calculated by first evaluating g(2) = 5.0000, then evaluating f(5.0000) = 6.0000.
Step-by-Step Solution:
- Given: f(x) = x + 1, g(x) = x^2 + 1
- Find: (f ∘ g)(2) = f(g(2))
- Step 1: Calculate g(2)
- g(2) = 2^2 + 1 = 5.0000
- Step 2: Calculate f(g(2)) = f(5.0000)
- f(5.0000) = 5.0000 + 1 = 6.0000
Composite Functions:
- • (f ∘ g)(x) = f(g(x)) - apply g first, then f
- • Order matters: f(g(x)) ≠ g(f(x)) in general
- • Domain: x must be in domain of g, and g(x) in domain of f
- • Chain rule: (f ∘ g)'(x) = f'(g(x)) · g'(x)
Quick Example Result
For f(x) = x + 1, g(x) = x² + 1, at x = 2:
f(g(2)) = f(5) = 6
Composite function: f(g(x)) = (x² + 1) + 1 = x² + 2
How This Calculator Works
Our composite function calculator applies fundamental principles of function composition from calculus and algebra. The calculator evaluates composite functionsby systematically applying one function to the output of another, following the mathematical definition of function composition.
Function Composition Formulas
(f ∘ g)(x) = f(g(x))1. Calculate g(x) → 2. Calculate f(g(x))x ∈ Domain(g) ∧ g(x) ∈ Domain(f)Function composition requires careful attention to order and domain restrictions. The notation f ∘ g is read as "f composed with g" and means applying g first, then f. The domain of the composite function is restricted by both component functions.
Shows the flow of input through g(x) then f(x) to produce f(g(x))
Mathematical Foundation
Function composition is a fundamental operation in mathematics that combines two or more functions to create a new function. This concept is essential in calculus (chain rule), algebra (function transformations), and many areas of applied mathematics. Understanding composition helps in analyzing complex relationships and solving multistep problems.
- Composition is associative: (f ∘ g) ∘ h = f ∘ (g ∘ h)
- Composition is generally not commutative: f ∘ g ≠ g ∘ f
- The identity function is the identity element for composition
- Domain restrictions must be carefully considered for composite functions
Sources & References
- Calculus: Early Transcendentals - James Stewart (9th Edition)Comprehensive treatment of function composition and chain rule
- Mathematical Association of America - Function Composition Teaching ResourcesEducational standards for teaching composite functions
- MIT OpenCourseWare - Single Variable CalculusAcademic materials on function composition and applications
Need help with other function calculations? Check out our derivative calculator and partial derivative calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Functions:
- g(x): x + 273.15 (Celsius to Kelvin)
- f(x): 1.8x (scaling factor)
- Input: x = 25°C
Composition Steps:
- Calculate g(25) = 25 + 273.15 = 298.15 K
- Calculate f(g(25)) = f(298.15)
- f(298.15) = 1.8 × 298.15 = 536.67
- Result: (f ∘ g)(25) = 536.67
Result: f(g(25)) = 536.67
This composite function first converts 25°C to Kelvin (298.15 K), then applies a scaling factor of 1.8 to get 536.67. This demonstrates how composite functions can model multi-step real-world processes.
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