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When dealing with polynomial functions, one common task is function evaluation. This involves calculating the function's value for specific inputs or "arguments." For instance, in our exercise, we have a function defined as \(g(x) = 3x^2\). To evaluate this function for a particular value of \(x\), we simply substitute that value into the formula.

• For \(g(-x)\), this means substituting \(-x\) into the formula wherever \(x\) appears, resulting in \(3(-x)^2\).
• In the case of \(g(2x)\), substitute \(2x\) for \(x\), giving us \(3(2x)^2\).
• Finally, for \(g(a + h)\), replace \(x\) with \(a + h\), leading to \(3(a + h)^2\).

As you can see, function evaluation is a straightforward process once you get the hang of substituting different inputs into the given function formula.

Substitution in functions is a critical concept, especially in the evaluation process. This means replacing the variable \(x\) in a function formula with another expression or number. For polynomial functions, this can transform the expression significantly, requiring careful handling of arithmetic operations.
When substituting a variable or expression into a function, always follow these steps:

• Identify the expression you need to substitute into the original function.
• Replace every occurrence of the variable in the function definition with the chosen expression.
• Perform the necessary arithmetic operations to simplify the expression.

Let's take an example from our exercise:

• For \(g(-x)\), replace \(x\) with \(-x\), resulting in \(3(-x)^2\). Simplifying gives \(3x^2\) since the square of a negative is positive.
• In \(g(2x)\), substitute \(2x\) for \(x\), yielding \(3(2x)^2 = 12x^2\), after simplifying.
• For \(g(a+h)\), replace \(x\) with \(a+h\), leading to \(3(a+h)^2\). Simplify using the binomial expansion to get \(3(a^2 + 2ah + h^2)\).

Understanding substitution helps make function evaluation more manageable and leads to accurate arithmetic results.

The binomial expansion is a valuable algebraic tool used to simplify expressions involving binomials raised to a power. When evaluating functions that feature polynomial expressions like \(g(a+h)\), binomial expansion comes into play. The general formula for expanding a binomial \((a+b)^n\) is:\[(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \ldots + b^n\]For example, to expand \((a+h)^2\), you apply the binomial expansion:

• Calculate \(a^2\).
• Add \(2ah\), which is twice the product of \(a\) and \(h\).
• Finally, add \(h^2\).

So, the expanded form of \((a + h)^2\) is \(a^2 + 2ah + h^2\). In our exercise, when substituting \(a+h\) into our function, we used this expansion to rewrite \(3(a + h)^2\) as \(3(a^2 + 2ah + h^2)\).
Using binomial expansion can simplify many mathematical problems and improve your understanding of polynomial expressions.