To understand the function [tex]\( f(x) = e^x + 2 \)[/tex], we will go through its properties and provide detailed steps on how to evaluate it.
### Step-by-Step Explanation:
1. Function Definition:
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) = e^x + 2
\][/tex]
Here, [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828, and [tex]\( x \)[/tex] is the input variable.
2. Exponential Component:
The term [tex]\( e^x \)[/tex] represents the exponential function. Exponential functions grow rapidly and their rate of growth increases as the input [tex]\( x \)[/tex] increases.
3. Constant Addition:
The term [tex]\( +2 \)[/tex] is a constant that is added to [tex]\( e^x \)[/tex]. This constant shifts the entire exponential graph of [tex]\( e^x \)[/tex] upward by 2 units.
4. Function Evaluation:
To evaluate the function at a specific point, say [tex]\( x = a \)[/tex], substitute [tex]\( a \)[/tex] into the function:
[tex]\[
f(a) = e^a + 2
\][/tex]
This means you calculate the exponential part [tex]\( e^a \)[/tex] first and then add 2 to it.
### Examples:
- Example 1: Evaluate [tex]\( f(0) \)[/tex].
[tex]\[
f(0) = e^0 + 2 = 1 + 2 = 3
\][/tex]
Therefore, [tex]\( f(0) = 3 \)[/tex].
- Example 2: Evaluate [tex]\( f(1) \)[/tex].
[tex]\[
f(1) = e^1 + 2 \approx 2.71828 + 2 = 4.71828
\][/tex]
Therefore, [tex]\( f(1) \approx 4.71828 \)[/tex].
- Example 3: Evaluate [tex]\( f(-1) \)[/tex].
[tex]\[
f(-1) = e^{-1} + 2 \approx 0.36788 + 2 = 2.36788
\][/tex]
Therefore, [tex]\( f(-1) \approx 2.36788 \)[/tex].
### Graphical Representation:
The graph of [tex]\( f(x) = e^x + 2 \)[/tex] looks like the graph of [tex]\( e^x \)[/tex] but shifted upward by 2 units. It starts from just above 2 (when [tex]\( x \)[/tex] is very negative), crosses the y-axis at 3 (when [tex]\( x = 0 \)[/tex]), and rises exponentially as [tex]\( x \)[/tex] increases.
### Domain and Range:
- Domain: The domain of [tex]\( f(x) \)[/tex] is all real numbers, [tex]\( x \in (-\infty, \infty) \)[/tex]. This is because there are no restrictions on the values that [tex]\( x \)[/tex] can take for the exponential function [tex]\( e^x \)[/tex].
- Range: The range of [tex]\( f(x) \)[/tex] is [tex]\( (2, \infty) \)[/tex]. As [tex]\( e^x \)[/tex] always yields positive values and approaches 0 as [tex]\( x \)[/tex] goes to negative infinity, [tex]\( e^x + 2 \)[/tex] will always be greater than 2.
In summary, the function [tex]\( f(x) = e^x + 2 \)[/tex] is an exponential function shifted upwards by 2 units, with a domain of all real numbers and a range of values greater than 2.
