To help you understand the function [tex]\( f(x) = 2x + e^{2 - 2x} \)[/tex], let's break it down and consider each component in detail.
### Step-by-Step Analysis:
1. Identifying the Function Components:
- The function [tex]\( f(x) \)[/tex] consists of two main components:
- A linear term: [tex]\( 2x \)[/tex]
- An exponential term: [tex]\( e^{2 - 2x} \)[/tex]
2. Understanding the Linear Term:
- The linear term [tex]\( 2x \)[/tex] is simply a line with a slope of 2. For any input [tex]\( x \)[/tex], it scales [tex]\( x \)[/tex] by 2.
3. Understanding the Exponential Term:
- The exponential term [tex]\( e^{2 - 2x} \)[/tex] involves the base [tex]\( e \)[/tex] (approximately 2.71828).
- Within the exponent, we have [tex]\( 2 - 2x \)[/tex].
- This term describes an exponential decay because the exponent [tex]\( -2x \)[/tex] decreases as [tex]\( x \)[/tex] increases.
4. Combining the Components:
- When these components are combined, [tex]\( f(x) = 2x + e^{2 - 2x} \)[/tex], the result is a function that combines linear growth (from [tex]\( 2x \)[/tex]) with exponential decay (from [tex]\( e^{2 - 2x} \)[/tex]).
### Step-by-Step Evaluation:
To evaluate [tex]\( f(x) \)[/tex] at any specific value of [tex]\( x \)[/tex], follow these steps:
1. Substitute the value of [tex]\( x \)[/tex] into the function.
2. Perform operations as per the order of operations (PEMDAS/BODMAS):
- Evaluate the exponent first.
- Calculate the exponential base [tex]\( e \)[/tex].
- Perform the multiplication and addition.
### Example Evaluations:
#### Example 1: Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]
1. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 2(0) + e^{2 - 2(0)}
\][/tex]
2. Simplify the exponent:
[tex]\[
e^{2 - 2(0)} = e^2
\][/tex]
3. Add the results:
[tex]\[
f(0) = 0 + e^2 = e^2
\][/tex]
#### Example 2: Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex]
1. Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[
f(1) = 2(1) + e^{2 - 2(1)}
\][/tex]
2. Simplify the exponent:
[tex]\[
e^{2 - 2(1)} = e^0
\][/tex]
3. Remember that [tex]\( e^0 = 1 \)[/tex]:
[tex]\[
f(1) = 2 + 1 = 3
\][/tex]
By understanding and following these steps, you can evaluate [tex]\( f(x) = 2x + e^{2 - 2x} \)[/tex] at any given value of [tex]\( x \)[/tex], and analyze the behavior of the function across different intervals.
