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To solve problems involving functions, a key technique is to substitute a given value into the function. This means we replace the variable, typically represented as 'x', with a specific number or symbol.
Let's walk through the process using the given exercise:
1. Identify the function: We are given the function \( g(x) = -x^2 + 4x + 1 \).
2. Recognize which value to substitute: In this case, we need to find \( g(e) \), so we substitute 'e' for 'x'.
3. Rewrite the function with the substituted value: After substituting 'e' into the function, we get \( g(e) = -e^2 + 4e + 1 \).
By following these steps, you can substitute any given value into a function to find the result efficiently. Understanding substitution is fundamental for working with functions and will be useful in a wide range of mathematical problems.

Function notation is a concise way to express functions and their evaluations. It uses letters like f, g, or h to name functions and parentheses to show the input value.
In our example, the function is given as \( g(x) = -x^2 + 4x + 1 \):
- Here, 'g' is the name of the function.
- 'x' inside the parentheses is the input, or the variable. When we write \( g(e) \), it simply means we're replacing 'x' with 'e' in the function.
- The right side of the equation represents the output after evaluating the function with the given input.
Using function notation helps in understanding and clearly communicating the operations being performed.
For example, \( g(e) = -e^2 + 4e + 1 \) succinctly conveys that we need to substitute 'e' into the function \( g(x) = -x^2 + 4x + 1 \). This notation is very useful, especially when dealing with more complex functions and their evaluations.

After substituting a value into a function, the next step is to simplify the resulting expression. Simplification involves combining like terms and performing arithmetic operations to make the expression as simple as possible.
Let's look at our example \( g(e) = -e^2 + 4e + 1 \):
- Identify any like terms. In this case, there are no like terms to combine because \( -e^2, 4e, \) and 1 are all distinct.
- Perform any necessary arithmetic operations, which in this case, are already simplified as it stands.
In some exercises, you might need to perform additional operations such as factoring or expanding terms. Always aim to express the final result in its simplest form to avoid errors and to make further calculations easier.
This step ensures that you clearly understand the result of the function after substitution and can confidently use it in any subsequent calculations.