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Function substitution is a method to evaluate a function at a different point or form.
We start with a given function, such as a polynomial or algebraic expression.
In our case, we have the function: \(g(x) = -x^2 + 4x + 1\). To find \(g(-x)\), we substitute \(-x\) for \(x\).
This substitution changes every instance of \(x\) in the original function to \(-x\). Here are some steps to follow for substituting variables in a function:

• Identify the variable to be replaced.
• Substitute the new variable or expression in place of the original variable.
• Simplify the resulting expression if necessary.

Now let's substitute \(-x\) into the given function:
\(g(-x) = -(-x)^2 + 4(-x) + 1\).
Next, we simplify the expression to find the new function.

Polynomial functions are algebraic expressions that include terms with variables raised to whole number powers.
They consist of coefficients and variables, and they can be of different degrees depending on the highest power of the variable.
For example, the function \(g(x) = -x^2 + 4x + 1\) is a second-degree polynomial because the highest power of \(x\) is 2.
The components of a polynomial function can be:

• Constant term (e.g., 1)
• Linear term (e.g., 4x extends the behavior linearly)
• Quadratic or higher-degree terms (e.g., \(-x^2\) represents the quadratic term)

Polynomials are valuable in calculus and algebra due to their simple differentiation and integration properties.
When substituting a value, each term involving \(x\) must be carefully rewritten according to the new variable or value.
This helps maintain the accuracy in calculations and transformations of the functions.

Simplifying expressions involves reducing a mathematical expression to its simplest form.
This often includes combining like terms, expanding or factoring expressions, and reducing fractions.
When we substituted \(-x\) into the function \(g(x)\), we formed \(g(-x) = -(-x)^2 + 4(-x) + 1\).
To simplify this:
First, recognize that \(-(-x)^2 = -x^2\), because \((-x)^2 = x^2\) and the negative sign outside makes it negative.
Next, simplify each term involving \(-x\):
\[ 4(-x) = -4x \ \ So \ g(-x)\ becomes \ -x^2 - 4x + 1 \].
Simplifying ensures that the function is expressed in its most efficient and understandable form.
This step is fundamental in solving equations, performing calculus operations, and understanding the behavior of mathematical functions.