Math Problem Statement
Describe the long run behavior of f(x) = 2(5)^x - 4.
Solution
The given function is , and you are asked to describe the long-run behavior, meaning what happens to as (positive infinity) and (negative infinity).
Long-run behavior as :
- When , the term grows extremely large because the base of the exponent (5) is greater than 1. This causes to also grow without bound.
- Therefore, as , the term dominates, and the function .
Long-run behavior as :
- When , the term approaches 0 because raising 5 to a negative power results in very small values (approaching zero).
- So, approaches zero as well. The constant term remains.
- Therefore, as , .
Would you like further details or have any questions?
Related Questions:
- What is the y-intercept of ?
- How would the long-run behavior change if the base of the exponent was less than 1?
- How does the value of the constant term affect the graph of the function?
- Can you find the horizontal asymptote of the function ?
- What is the derivative of , 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 or .
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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