91Ó°ÊÓ

In algebra, functions are like special machines that take an input (usually a number), perform some operations on it, and give an output. Function evaluation is the process of finding the output of a function for a specific input.
For example, consider the function definition given as \(g(x) = -x^2 + 4x + 1\). In this definition, \(x\) is the input variable.
Function evaluation involves these steps:

• Identifying the given function
• Substituting the specific input value into the function
• Performing the calculations to get the output

In our exercise, we need to find \(g(0)\). The steps are:
1. Identify the function: \(g(x) = -x^2 + 4x + 1\)
2. Substitute \(x = 0\) into the function: \(g(0) = -0^2 + 4(0) + 1\)
3. Simplify the resulting expression to get the result.
This is an essential process in algebra and helps to understand how functions work.

The substitution method is a fundamental technique used in algebra to solve equations and evaluate functions. It involves replacing a variable with a given value to simplify and solve the equation.
To use the substitution method:

• Identify the variable in the function
• Replace the variable with the given value

In our example, we substitute \(x = 0\) into the function \(g(x) = -x^2 + 4x + 1\). This means that we replace every occurrence of \(x\) with 0:

• Initial function: \(g(x) = -x^2 + 4x + 1\)
• After substitution: \(g(0) = -(0)^2 + 4(0) + 1\)

This method helps simplify and make calculations straightforward. Once the substitution is done, the next step is to simplify the resulting expression.

Simplifying expressions means reducing them to their simplest form. This involves performing arithmetic operations and combining like terms.
After substitution, we have the expression \(g(0) = -(0)^2 + 4(0) + 1\). We simplify by:

• Calculating powers and multiplication: \(-(0)^2\) becomes 0 and \(4(0)\) becomes 0
• Adding the results: \(g(0) = 0 + 0 + 1\)

So, the simplified expression is \(g(0) = 1\).
Simplifying makes it easier to see the result more clearly. Understanding how to simplify expressions is crucial for solving algebraic problems efficiently.