Math Problem Statement

Describe the long run behavior of f(x) = 2(5)^x - 4.

Solution

The given function is f(x)=2(5)x4f(x) = 2(5)^x - 4, and you are asked to describe the long-run behavior, meaning what happens to f(x)f(x) as xx \to \infty (positive infinity) and xx \to -\infty (negative infinity).

Long-run behavior as xx \to \infty:

  1. When xx \to \infty, the term 5x5^x grows extremely large because the base of the exponent (5) is greater than 1. This causes 2(5)x2(5)^x to also grow without bound.
  2. Therefore, as xx \to \infty, the term 2(5)x2(5)^x dominates, and the function f(x)f(x) \to \infty.

limxf(x)=\boxed{\lim_{x \to \infty} f(x) = \infty}

Long-run behavior as xx \to -\infty:

  1. When xx \to -\infty, the term 5x5^x approaches 0 because raising 5 to a negative power results in very small values (approaching zero).
  2. So, 2(5)x2(5)^x approaches zero as well. The constant term 4-4 remains.
  3. Therefore, as xx \to -\infty, f(x)4f(x) \to -4.

limxf(x)=4\boxed{\lim_{x \to -\infty} f(x) = -4}

Would you like further details or have any questions?

Related Questions:

  1. What is the y-intercept of f(x)=2(5)x4f(x) = 2(5)^x - 4?
  2. How would the long-run behavior change if the base of the exponent was less than 1?
  3. How does the value of the constant term 4-4 affect the graph of the function?
  4. Can you find the horizontal asymptote of the function f(x)=2(5)x4f(x) = 2(5)^x - 4?
  5. What is the derivative of f(x)=2(5)x4f(x) = 2(5)^x - 4, and what does it tell us about the function’s behavior?

Tip:

When analyzing exponential functions, the base of the exponential term is critical in determining whether the function grows or decays as xx \to \infty or xx \to -\infty.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Limits
Long-Run Behavior

Formulas

Limit of exponential functions as x approaches infinity or negative infinity

Theorems

Limit Theorem for Exponential Growth

Suitable Grade Level

Grades 10-12