Problem 14
Question
Describe the end behavior of \(g(x)=e^{-2 x}\).
Step-by-Step Solution
Verified Answer
#tag_title# Short Answer
#tag_content# The end behavior of the function g(x) = e^{-2x} is as follows: as x approaches positive infinity, g(x) approaches 0, and as x approaches negative infinity, g(x) approaches infinity.
1Step 1: Identify the function
First, we need to identify the function given:
$$g(x) = e^{-2x}$$
This function represents an exponential decay because e is raised to a negative power.
2Step 2: Find the limit as x approaches positive infinity
We want to find the limit of g(x) as x approaches positive infinity. Mathematically, we write:
$$\lim_{x \to \infty} e^{-2x}$$
Since we have an exponential function with a negative power, as x becomes larger, the exponent becomes more negative. This means that g(x) approaches zero.
$$\lim_{x \to \infty} e^{-2x} = 0$$
3Step 3: Find the limit as x approaches negative infinity
Now, let's find the limit of g(x) as x approaches negative infinity. Mathematically, we write:
$$\lim_{x \to -\infty} e^{-2x}$$
In this case, as x becomes more negative, the exponent becomes less negative or more positive. This means that g(x) will approach infinity.
$$\lim_{x \to -\infty} e^{-2x} = \infty$$
4Step 4: Describe the end behavior of g(x)
Now that we have found the limits as x approaches positive and negative infinity, we can describe the end behavior of g(x). As x approaches positive infinity, the function g(x) approaches 0. As x approaches negative infinity, the function g(x) approaches infinity.
Key Concepts
Exponential DecayLimit as x approaches infinityExponential Functions
Exponential Decay
Exponential decay refers to the process where quantities reduce at a consistent rate over time. This phenomenon is often seen in various natural processes and mathematical models. For the function \(g(x) = e^{-2x}\), we have an exponential decay because the exponent is negative.
Let's break it down: in an exponential decay scenario, as the independent variable increases (like time or distance), the value of the function decreases continuously. This is typical in scenarios like radioactive decay or depreciation of assets where values diminish rapidly at first and then slow down as they approach zero.
For an exponential decay function, the rate at which a decrease happens is proportional to its current value. This is what makes it quite powerful and commonly used in modeling real-world situations.
Let's break it down: in an exponential decay scenario, as the independent variable increases (like time or distance), the value of the function decreases continuously. This is typical in scenarios like radioactive decay or depreciation of assets where values diminish rapidly at first and then slow down as they approach zero.
For an exponential decay function, the rate at which a decrease happens is proportional to its current value. This is what makes it quite powerful and commonly used in modeling real-world situations.
Limit as x approaches infinity
Limits are a fundamental concept in calculus, providing insights into the behavior of functions as variables head toward specific points or toward infinity. When we talk about the "limit as \(x\) approaches infinity," we essentially want to know what value, if any, a function settles into as \(x\) becomes extremely large.
For \(g(x) = e^{-2x}\), the limit as \(x\) approaches positive infinity can be calculated as follows:
\[ \lim_{x \to \infty} e^{-2x} = 0 \] This limit tells us that the function \(g(x)\) effectively gets closer and closer to zero but never really reaches it, showcasing characteristic end behavior for exponential decay functions.
For \(g(x) = e^{-2x}\), the limit as \(x\) approaches positive infinity can be calculated as follows:
- The exponent \(-2x\) becomes increasingly negative as \(x\) grows larger.
- As a result, \(e^{-2x}\) approaches zero since the base of an exponential function raised to a large negative power leads the function's value closer to 0.
\[ \lim_{x \to \infty} e^{-2x} = 0 \] This limit tells us that the function \(g(x)\) effectively gets closer and closer to zero but never really reaches it, showcasing characteristic end behavior for exponential decay functions.
Exponential Functions
Exponential functions are a cornerstone of advanced mathematics, representing processes that grow or decay at a constant rate. These functions are characterized by their unique form, \(f(x) = a \, e^{bx}\), where several components define its behavior:
- \(a\) is a constant coefficient that scales the function up or down.
- \(e\) is the base of the natural logarithm, an irrational number approximately equal to 2.71828.
- \(b\) dictates the rate of growth (if positive) or decay (if negative).
- The negative exponent, \(-2x\), implies a rapid decay as \(x\) rises.
- Exponential functions like this are key to modeling decay processes that go from a higher initial value towards zero.
- They exhibit unique properties, such as compositional invertibility and aligning with the growth-and-decay rule of "same shape at any stage," which makes them predictable and useful in calculations.
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