The graph of an exponential function is given. Select the function for each graph from the following options:
$f(x) = 3^x, \quad g(x) = 3^{x-1}, \quad h(x) = 3^x - 1, \\f(x) = -3^x, \quad G(x) = 3^{-x}, \quad H(x) = -3^{-x}.$

Verified step by step guidance

1. Step 1: Identify the horizontal asymptote of the graph. The graph shows a horizontal asymptote at \(y = -1\), which means the function approaches \(-1\) as \(x\) goes to negative infinity.
2. Step 2: Recall that the basic exponential function \$3^x\( has a horizontal asymptote at \)y = 0\(. To shift this asymptote to \)y = -1\(, the function must be vertically shifted down by 1 unit. This suggests the function has the form \)3^x - 1$.
3. Step 3: Check the point given on the graph, which is \((1, 5)\). Substitute \(x = 1\) into the candidate function \$3^x - 1\( to verify: \)3^1 - 1 = 3 - 1 = 2\(, which does not match the point \)(1, 5)$.
4. Step 4: Consider the function \$3^{x-1}\(, which shifts the graph horizontally to the right by 1 unit. The asymptote remains at \)y = 0\(, so this does not match the asymptote at \)y = -1$.
5. Step 5: Since the asymptote is at \(y = -1\) and the graph passes through \((1, 5)\), the function must be \(f(x) = 3^x - 1\). The discrepancy in step 3 suggests rechecking the point or considering the vertical shift as the key feature to identify the function.

Key Concepts

Exponential Functions and Their Graphs

Exponential functions have the form f(x) = a^x, where the base a is positive and not equal to 1. Their graphs show rapid growth or decay, depending on the base and the exponent's sign. Key features include a horizontal asymptote and passing through specific points like (0,1) for f(x) = a^x.

Transformations of Exponential Functions

Transformations such as shifts and reflections modify the graph of an exponential function. Horizontal shifts change the input (e.g., f(x) = 3^(x-1) shifts right by 1), vertical shifts add or subtract constants (e.g., f(x) = 3^x - 1 shifts down by 1), and reflections flip the graph across axes.

Horizontal Asymptotes in Exponential Functions

Exponential functions have horizontal asymptotes that the graph approaches but never touches. The asymptote is typically y=0 for basic functions, but vertical shifts move it up or down. In the given graph, the asymptote at y = -1 indicates a vertical shift downward by 1.