Exponential Function Calculator
Evaluate exponential functions or solve for parameters from points
Exponential Function Calculator
Current Function: f(x) = a × b^x
Mode: Evaluate function at x value
Result
Function Value
f(2) = 18.000000
Function
f(x) = 2 × 3^x
At x = 2
y = 18.000000
Explanation
Evaluating the exponential function f(x) = 2 × 3^x at x = 2
Step-by-Step Solution
1.f(x) = 2 × 3^x
2.f(2) = 2 × 3^2
3.f(2) = 2 × 9.000000
4.f(2) = 18.000000
Function Forms
Basic Exponential
f(x) = b^x
Scaled Exponential
f(x) = a × b^x
Natural Exponential
f(x) = e^(cx)
Scaled Natural
f(x) = a × e^(cx)
Key Properties
•
Growth/Decay
b > 1: growth, 0 < b < 1: decay
•
Always Positive
f(x) > 0 for all x
•
Horizontal Asymptote
y = 0 (or shifted)
•
Euler's Number
e ≈ 2.71828 (natural base)
Understanding Exponential Functions
What is an Exponential Function?
An exponential function is a mathematical function where the variable appears as an exponent. The most basic form is f(x) = b^x, where b is the base and x is the exponent.
Key Characteristics
- •Always positive output values
- •Continuous and smooth curve
- •Rapid growth or decay behavior
- •Horizontal asymptote at y = 0
Common Forms
Basic: f(x) = b^x
Base b raised to power x
Example: f(x) = 2^x
Scaled: f(x) = a × b^x
Scaled by factor a
Example: f(x) = 3 × 2^x
Natural: f(x) = e^(cx)
Base e with coefficient c
Example: f(x) = e^(0.5x)
Real-World Applications
Population Growth
P(t) = P₀ × e^(rt) where r is growth rate
Compound Interest
A(t) = P × (1 + r)^t for annual compounding
Radioactive Decay
N(t) = N₀ × e^(-λt) where λ is decay constant