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Use the given function \( f \) to answer parts (a) through (f) below. \[ f(x)=e^{x+2}+1 \]

Ask by Weber Crawford. in the United States
Mar 18,2025

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**(a) Domain:** All real numbers **(b) Range:** \( (1, \infty) \) **(c) \( x \)-intercepts:** None **(d) \( y \)-intercept:** \( (0, e^2 + 1) \) **(e) Inverse Function:** \( f^{-1}(x) = \ln(x - 1) - 2 \) **(f) Transformations:** Shifted 2 units left and 1 unit up from \( e^x \).

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Mind Expander

Did you know that the function \( f(x) = e^{x+2} + 1 \) can be related to real-life growth patterns? The exponential function, represented by \( e^x \), is often used to model things like population growth, radioactive decay, and even compound interest in finance! As \( x \) increases, the term \( e^{x+2} \) soars, demonstrating how quickly certain processes can escalate. Now, let’s dive into some common pitfalls when dealing with exponential functions! A mistake often made is forgetting the base of the exponential or miscalculating when plugging in values of \( x \). Make sure to double-check your calculations and remember the properties of exponents, like \( e^{a+b} = e^a \cdot e^b \), to simplify your work and avoid errors.

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