Questions: Graph the function f(x)=-5^(x+5). Give the domain and range.

Transcript text: Graph the function $f(x)=-5^{x+5}$. Give the domain and range.

Solution

Solution Steps

Step 1: Identify the function

The function given is f(x) = -5^x + 5.

Step 2: Find the y-intercept

The y-intercept occurs when x = 0. f(0) = -5⁰ + 5 f(0) = -1 + 5 f(0) = 4 The y-intercept is (0, 4).

Step 3: Find the horizontal asymptote

As x approaches infinity, the term -5^x approaches negative infinity. Adding 5 to a very large negative number results in a value approaching negative infinity. Therefore, there is no horizontal asymptote in the positive x direction.

As x approaches negative infinity, the term -5^x approaches 0. Thus, the function approaches -0 + 5, or 5. The horizontal asymptote is y = 5.

Final Answer:

The graph of the function f(x) = -5^x + 5 has a y-intercept at (0, 4) and a horizontal asymptote at y = 5. The graph decreases exponentially as x increases, approaching negative infinity. As x decreases, the graph approaches the horizontal asymptote y=5. The domain is all real numbers, and the range is (-∞, 5).

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