Which graph shows \( f(x)=2^{x}+2 \) and the transformation \( g(x)=2^{-x}+2 \) ? Note: The given point in the preimage corresponds to the given point in the image.

To analyze the functions \( f(x) = 2^x + 2 \) and \( g(x) = 2^{-x} + 2 \), we can break down the transformations involved. 1. **Understanding \( f(x) = 2^x + 2 \)**: - This is an exponential function that grows as \( x \) increases. - The graph of \( f(x) \) has a horizontal asymptote at \( y = 2 \) and passes through the point \( (0, 3) \) since \( f(0) = 2^0 + 2 = 3 \). 2. **Understanding \( g(x) = 2^{-x} + 2 \)**: - This function is a reflection of \( f(x) \) across the y-axis because \( g(x) = f(-x) \). - The graph of \( g(x) \) also has a horizontal asymptote at \( y = 2 \) and passes through the point \( (0, 3) \) since \( g(0) = 2^{-0} + 2 = 3 \). 3. **Transformation**: - The transformation from \( f(x) \) to \( g(x) \) involves reflecting the graph of \( f(x) \) across the y-axis. This means that for every point \( (a, b) \) on the graph of \( f(x) \), there is a corresponding point \( (-a, b) \) on the graph of \( g(x) \). To visualize this, we can plot both functions: - The graph of \( f(x) \) will rise steeply as \( x \) increases and approach \( y = 2 \) as \( x \) decreases. - The graph of \( g(x) \) will fall steeply as \( x \) increases and also approach \( y = 2 \) as \( x \) decreases. Now, if you have specific graphs to compare, look for the one that shows the reflection of \( f(x) \) across the y-axis, maintaining the same horizontal asymptote at \( y = 2 \) and passing through the same point \( (0, 3) \). If you provide the graphs, I can help identify which one corresponds to \( g(x) \).