Exponential Function Calculator

Evaluate f(x) = a · bx. Determine growth vs. decay, find the y-intercept, asymptote, and see a table of values.

Define Your Exponential Function

f(x) = 2 * 3^x

Result

f(x) =
162
Function ---
Classification ---
Y-Intercept (x=0) ---
Horizontal Asymptote ---
Domain ---
Range ---
Growth/Decay Factor ---
Growth/Decay Rate ---

Step-by-Step Evaluation

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Understanding Exponential Functions

An exponential function is a mathematical function of the form f(x) = a · bx, where a is the initial value (coefficient), b is the base (growth or decay factor), and x is the exponent. The base b must be positive and not equal to 1 for the function to be truly exponential.

Exponential functions model many real-world phenomena: population growth, radioactive decay, compound interest, the spread of disease, and the cooling of hot objects.

Key Properties

Exponential Growth (b > 1)

When the base is greater than 1, the function increases rapidly as x increases.

f(x) = 2 * 3^x (grows)

Exponential Decay (0 < b < 1)

When the base is between 0 and 1, the function decreases as x increases.

f(x) = 100 * 0.5^x (decays)

Y-Intercept

The y-intercept is always at (0, a) since b^0 = 1.

f(0) = a * b^0 = a

Horizontal Asymptote

The horizontal asymptote is y = 0 (the x-axis) for standard exponential functions.

As x -> -inf, f(x) -> 0 (if b > 1)

Growth Rate and Factor

The growth factor is the base b. The growth rate is (b - 1) expressed as a percentage. For example, if b = 1.05, the growth rate is 5%. If b = 0.9, the decay rate is 10%.

Domain and Range

  • Domain: All real numbers (-infinity, +infinity) -- you can plug any x value into an exponential function.
  • Range: If a > 0, the range is (0, +infinity). If a < 0, the range is (-infinity, 0).

Applications of Exponential Functions

  • Population Growth: P(t) = P0 · (1 + r)t
  • Compound Interest: A = P(1 + r/n)nt
  • Radioactive Decay: N(t) = N0 · (1/2)t/h
  • Bacterial Growth: B(t) = B0 · 2t/d (doubling every d hours)