- The function \(f(x) = 2^{x-3}\) is an exponential growth function because the base \(2\) is greater than 1.
- As \(x \to \infty\), the value of \(2^{x-3}\) increases without bound toward \(\infty\).
- As \(x \to -\infty\), the value of \(2^{x-3}\) approaches 0, which is the function's horizontal asymptote.
- Based on this analysis, the statement that correctly describes the end behavior is: For very high \(x\)-values, \(f(x)\) moves toward positive infinity.
- Final Answer: \(\boxed{\text{For very high } x\text{-values, } f(x) \text{ moves toward positive infinity.}}\)
Explanation
- Analyzing the Function Behavior
The function provided is \(f(x) = 2^{x-3}\). This is an exponential function of the form \(f(x) = a \cdot b^x\), where the base \(b = 2\) is greater than 1. Exponential functions with a base greater than 1 exhibit exponential growth, meaning the function values increase as \(x\) increases. To determine the end behavior, we evaluate the function at extreme values of \(x\), specifically as \(x\) approaches positive infinity and negative infinity.
- Evaluating Limit as x Approaches Infinity
First, let us examine the behavior as \(x\) becomes very large (i.e., as \(x \to \infty\)). The expression is \(f(x) = 2^{x-3}\). As \(x\) increases without bound, the exponent \(x-3\) also increases without bound. Consequently, \(2^{x-3}\) grows larger and larger, approaching positive infinity. Mathematically, \(\lim_{x \to \infty} 2^{x-3} = \infty\). This indicates that for very high \(x\)-values, the function moves toward positive infinity.
- Evaluating Limit as x Approaches Negative Infinity
Next, let us examine the behavior as \(x\) becomes very small (i.e., as \(x \to -\infty\)). As \(x\) decreases, the exponent \(x-3\) becomes a large negative number. We can rewrite the function as \(f(x) = \frac{2^x}{2^3} = \frac{1}{8} \cdot 2^x\). As \(x \to -\infty\), \(2^x\) approaches 0. Therefore, \(\lim_{x \to -\infty} \frac{1}{8} \cdot 2^x = 0\). This means the function approaches the horizontal asymptote at \(y = 0\) as \(x\) moves toward negative infinity.
- Determining the Correct Statement
Comparing our findings with the given choices:
- "For very high \(x\)-values, \(f(x)\) moves toward positive infinity" - This matches our calculation from Step 2.
- "For very high \(x\)-values, \(f(x)\) moves toward negative infinity" - Incorrect.
- "For very high \(x\)-values, \(f(x)\) moves toward the horizontal asymptote" - Incorrect, as the function grows towards infinity in this direction.
- "For very low \(x\)-values, \(f(x)\) moves toward negative infinity" - Incorrect, as it approaches 0.
Thus, the correct statement is that for very high \(x\)-values, \(f(x)\) moves toward positive infinity.
Examples
Exponential functions are vital in modeling many real-world scenarios, such as the growth of bacteria or compound interest. For instance, if you invest money in a savings account that compounds continuously, the amount of money in the account can be modeled by an exponential function like \(A(t) = P \cdot e^{rt}\). Understanding the end behavior of these functions helps you predict the long-term value of your investment: just as the function \(f(x) = 2^{x-3}\) grows without bound as time (or \(x\)) increases, a well-invested sum of money will grow significantly over a long period. Conversely, recognizing that the function approaches zero as \(x\) becomes very small helps understand how quickly a value decays when time is reversed or when modeling radioactive decay processes.