Question

what is the range of exponential function g?

Explanation:

Step1: Recall Exponential Function Range

Exponential functions have the form \( y = a^x + k \). The range of a basic exponential function \( y = a^x \) (where \( a>0, a
eq1 \)) is \( y > 0 \) if \( a > 1 \) or \( a < 1 \) (since it approaches 0 but never touches it). If there's a vertical shift \( k \), the range shifts. From the graph, the function seems to have a horizontal asymptote. Looking at the graph, the function is above a certain line. Wait, maybe the function here is a decaying exponential? Wait, the graph shown: let's check the y - intercept. Wait, the graph is in the coordinate plane. Wait, maybe the function is \( y = b^x + c \). Wait, the key is that for exponential functions, the range is all real numbers greater than the horizontal asymptote. From the graph, the horizontal asymptote seems to be \( y=- 2 \)? Wait, no, wait the graph: looking at the axes, the y - axis is on the right? Wait, maybe the graph is a bit rotated? Wait, no, the standard exponential function: if it's a decaying exponential, like \( y = (\frac{1}{a})^x + k \), the range is \( y>k \). Wait, from the graph, the function is above \( y = - 2 \)? Wait, no, maybe I misread. Wait, the graph: the line (the exponential curve) is above \( y=-2 \)? Wait, no, let's re - examine. Wait, the graph is drawn such that as \( x \) approaches negative infinity, the function goes up, and as \( x \) approaches positive infinity, it approaches a horizontal line. Wait, the horizontal asymptote: looking at the graph, the function approaches \( y = - 2 \) from above? Wait, no, maybe the asymptote is \( y=-2 \), so the range is \( y > - 2 \). Wait, but let's think again. The general form of an exponential function is \( y = ab^x + k \). The range of \( y = ab^x + k \) is \( y>k \) if \( a>0 \) (since \( ab^x>0 \) when \( a > 0 \), so \( ab^x + k>k \)) or \( y < k \) if \( a < 0 \), but \( a>0 \) for exponential functions (usually, since if \( a<0 \), it's not a real - valued function for all real \( x \)). So from the graph, the horizontal asymptote is \( y=-2 \), so the range is all real numbers greater than \( - 2 \), i.e., \( y > - 2 \), or in interval notation \( (-2,\infty) \).

Step2: Confirm from Graph

Looking at the graph, the function never goes below \( y=-2 \) (the horizontal asymptote) and takes all values above it. So the range is all real numbers greater than \( - 2 \).

Answer:

The range of the exponential function \( g \) is all real numbers greater than \( - 2 \), or in interval notation \( (-2,\infty) \)