Draw the graph of the function and state the domain and range. \( f(x)=5^{x}+3 \)

We are given the function   f(x) = 5ˣ + 3 Step 1. Domain and Range • Domain:  Since the exponential function 5ˣ is defined for all real numbers, the domain is   (–∞, ∞). • Range:  The basic function 5ˣ has a range (0, ∞). Adding 3 shifts every output upward, so the new outputs are always greater than 3. Although f(x) approaches 3 as x → –∞, it never actually equals 3.  Thus, the range is   (3, ∞). Step 2. Graph Description • Key Characteristics:  – The graph is an exponential curve that is increasing because 5ˣ increases as x increases.  – It has a horizontal asymptote at y = 3 (since as x → –∞, 5ˣ → 0 and f(x) → 0 + 3 = 3).  – The y-intercept occurs when x = 0:   f(0) = 5⁰ + 3 = 1 + 3 = 4, giving the point (0, 4). • Plot Additional Points:  – At x = 1: f(1) = 5¹ + 3 = 5 + 3 = 8 → (1, 8)  – At x = –1: f(–1) = 5^(–1) + 3 = 1/5 + 3 = 3.2 → (–1, 3.2) • Graph Sketching Guidelines:  1. Draw a horizontal dashed line at y = 3 to represent the asymptote.  2. Plot the point (0, 4) on the y-axis.  3. Plot the points (1, 8) and (–1, 3.2) to help shape the curve.  4. For x > 0, the curve rises rapidly.  5. For x < 0, it gets closer and closer to the line y = 3 without touching it. When you sketch the graph: - Start near the line y = 3 on the far left and gradually curve upward as x increases, - Emphasize that the function never reaches y = 3, - And clearly show the rapid increase for positive values of x. Step 3. Final Answer • Domain: (–∞, ∞) • Range: (3, ∞) This completes the analysis of the function f(x) = 5ˣ + 3.