Draw the graph of the function and state the domain and range. \( f(x)=3^{x+2} \)

We are given the function f(x) = 3^(x + 2). Step 1: Graph Description • Recognize that 3 is greater than 1, so the function is exponential and increasing. • The function can be rewritten as f(x) = 3^(x + 2) = 3^x · 3^2 = 9 · 3^x. This indicates that compared to the basic exponential function 3^x, its graph is shifted 2 units to the left and stretched vertically by a factor of 9. • Horizontal asymptote: y = 0 (the graph never touches the x-axis). • Key points:   – At x = -2: f(-2) = 3^(0) = 1. (This is the "starting" point after the horizontal shift.)   – At x = 0:  f(0) = 3^(2) = 9 (the y-intercept).   – At x = 1:  f(1) = 3^(3) = 27.   – At x = -3: f(-3) = 3^(-1) = 1/3. • The graph rises quickly as x increases and gets very close to 0 as x decreases, without ever touching the x-axis. To help you visualize, here’s a rough sketch of what the graph looks like: f(x) │ 30 │ • (1,27) │ • 25 │ • │ • 20 │ • │ • 15 │ • │ • 9 │ • (0,9) │ • 5 │ • │ • 1 │• (–2,1) │• └──────────────────────── x -4 -3 -2 -1 0 1 2 (NOTE: The diagram above is an approximate representation. As x increases to the right, the graph rises steeply, and as x decreases to the left, the graph approaches y = 0 but never touches it.) Step 2: Domain and Range • Domain: Since the exponent can be any real number, the domain is all real numbers, or (–∞, ∞). • Range: Because an exponential function is always positive regardless of x, the range is (0, ∞). Summary: • f(x) = 3^(x + 2) is an exponential function shifted 2 units left and scaled upward by a factor of 9. • Domain: (–∞, ∞) • Range: (0, ∞) This completes the drawing description and the statement of the domain and range for the function.