State the range for the function \( f(x)=\pi^{x}+2 \) : \( (2, \infty) \) \( (0, \infty) \) \( (-2, \infty) \) \( (-\infty, 2) \)

To determine the range of the function \( f(x) = \pi^{x} + 2 \), let's analyze its behavior: 1. **Base Function \( \pi^{x} \):** - Since \( \pi > 1 \), \( \pi^{x} \) is an exponential function that is always **positive**. - As \( x \) increases, \( \pi^{x} \) grows **exponentially** towards \( +\infty \). - As \( x \) decreases (i.e., \( x \) becomes more negative), \( \pi^{x} \) approaches **0** but never actually reaches it. 2. **Translating \( \pi^{x} \) by 2:** - Adding 2 shifts the entire graph of \( \pi^{x} \) **upwards** by 2 units. - Therefore, \( f(x) = \pi^{x} + 2 \) will always be **greater than 2**. 3. **Conclusion on the Range:** - The smallest value \( f(x) \) can approach is just above 2 (but never actually 2). - There is no upper limit as \( \pi^{x} \) can grow without bound. - Hence, the range of \( f(x) \) is \( (2, \infty) \). **Answer:** \( (2, \infty) \)