Question

In Exercises \(1-4,\) graph the function. State its domain and range. $$y=e^{x}+3$$

Short answer

The function \(y=e^{x}+3\) is drawn as an upward shift of the base function \(e^x\) by 3 units. The domain of the function is all real numbers (\(-\infty \leq x \leq \infty\)). The range is all real numbers greater than 3 (\(3 < y \leq \infty\)).

Step-by-step solution

Understanding the given function

Analyze the given function, \(y=e^{x}+3\). This function is a transformation of the base function \(e^x\), where the base function is shifted upward by 3 units. Hence, it inherits characteristics of the base function \(e^x\). The function \(y=e^x\) always gives positive output, and it tends to infinity as x tends towards infinity. Conversely, it tends to zero but never reaches zero as x tends towards minus infinity.

Graphing the function

Since the given function is an upward shift of the base function, the graph will start from (0,3) and grow upward as x increases while it will approach but never reach the line \(y=3\) as x tends towards minus infinity. It will pass through the points (0,4), (1,e+3) because when we substitute x as 0, we get \(y=1+3=4\) and when we substitute x as 1 we get \(y=e+3\). Thus, the graph represents an increasing function.

Determining the Domain

The domain of the function \(y=e^{x}+3\) is all real numbers. This is because there are no restrictions on the values of x for \(e^x\), or for any upward or downward shift of \(e^x\). Hence, the domain is \(-\infty \leq x \leq \infty\).

Determining the Range

The range of the function \(y=e^{x}+3\) is all real numbers greater than 3. Since \(e^x\) is always positive and since as x tends towards minus infinity, \(e^x\) tends towards zero, so the minimum value of the given function is 3. Therefore, the range is \(3 < y \leq \infty\).