Questions: Use the function below to answer the following questions. k(x)=e^x+4 (a) Use transformations of the graph of y=e^x to graph the given function. (b) Write the domain and range in interval notation. (c) Write an equation of the asymptote.

Use the function below to answer the following questions.
k(x)=e^x+4
(a) Use transformations of the graph of y=e^x to graph the given function.
(b) Write the domain and range in interval notation.
(c) Write an equation of the asymptote.

Transcript text: Use the function below to answer the following questions. \[ k(x)=e^{x}+4 \] (a) Use transformations of the graph of $y=e^{x}$ to graph the given function. (b) Write the domain and range in interval notation. (c) Write an equation of the asymptote.

Solution

Solution Steps

Step 1: Solve for the graph transformation

The given function is $ k(x) = e^x + 4 $. This is a vertical shift of the graph $ y = e^x $ by 4 units upwards.

Step 2: Determine the domain and range

The domain of $ k(x) = e^x + 4 $ is all real numbers, $ (-\infty, \infty) $. The range of $ k(x) = e^x + 4 $ is $ (4, \infty) $ because $ e^x $ is always positive and adding 4 shifts it up.

Step 3: Find the equation of the asymptote

The horizontal asymptote of $ k(x) = e^x + 4 $ is $ y = 4 $.

Final Answer

(a) The graph of $ k(x) = e^x + 4 $ is a vertical shift of $ y = e^x $ by 4 units upwards.

(b) Domain: $ (-\infty, \infty) $ Range: $ (4, \infty) $

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