An exponential function \( f(x) = 2^x \) is given. Find \( f(3) \) and \( f(-2) \).
["Understanding the Exponential Function: Evaluating ( f(3) ) and ( f(-2) ) for ( f(x) = 2^x )", "The exponential function ( f(x) = 2^x ) is a fundamental mathematical model widely used in science, finance, computer science, and engineering. Its growth or decay behavior depends on the base and exponent, making it a powerful tool for describing phenomena like population growth, compound interest, and radioactive decay.", "In this article, we explore the exponential function ( f(x) = 2^x ), focusing on calculating two key values:\n- ( f(3) )\n- ( f(-2) )", "Understanding how to evaluate this function helps build a solid foundation in exponential mathematics.", "---", "What is ( f(x) = 2^x )?", "The function ( f(x) = 2^x ) represents exponential growth with base 2. When ( x ) is a positive integer, ( f(x) ) grows rapidly. For example:", "- ( f(0) = 2^0 = 1 )\n- ( f(1) = 2^1 = 2 )\n- ( f(2) = 2^2 = 4 )\n- ( f(3) = 2^3 = 8 ) — a clear example of rapid escalation", "But what does ( f(3) ) really mean?", "### Calculating ( f(3) = 2^3 )", "Using the basic rule for exponents:", "[\nf(3) = 2^3 = 2 \ imes 2 \ imes 2 = 8\n]", "So, when the input is 3, the output is 8 — a result that grows noticeably fast compared to linear functions.", "---", "Exploring Negative Exponents: Finding ( f(-2) )", "What happens when the input is negative? Specifically:", "[\nf(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}\n]", "This shows a key property of exponential functions: ( a^{-x} = \frac{1}{a^x} ). Negative exponents represent reciprocals, leading to fractional values less than 1.", "Even with a negative exponent, ( f(-2) = 0.25 ), which is still positive but much smaller than 1.", "---", "Real-World Significance", "Exponential functions model growth processes. Here’s where ( f(x) = 2^x ) applies:", "- Finance: The doubling of money under compound interest follows exponential patterns.\n- Biology: Bacteria doubling every hour can be modeled as ( 2^x ).\n- Physics: Radioactive decay, though usually decay (( a^x ) with ( 0 < a < 1 )), showcases how exponents describe change over time.", "Calculating ( f(3) = 8 ) illustrates a quantity increasing 8-fold in just three units — powerful for prediction and planning.", "---", "Conclusion", "Evaluating the exponential function ( f(x) = 2^x ) at key points reveals its dynamic behavior:", "- ( f(3) = \boxed{8} )\n- ( f(-2) = \boxed{\frac{1}{4}} )", "Whether used in problem-solving, data analysis, or scientific modeling, mastering such functions equips learners to understand and harness exponential change in real-world contexts. Understanding how to compute ( 2^x ) for positive and negative values builds confidence in tackling more complex exponential models.", "Keywords: exponential function, ( f(x) = 2^x ), evaluate ( f(3) ), evaluate ( f(-2) ), exponential growth, mathematics education, real-world applications, negative exponents."]
