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Evaluating functions is a fundamental concept in algebra and calculus. It involves finding the output of a function for a given input. Here's how you can do it:

First, understand the function you're working with. In our example, it's given as: \(f(x) = -3x + 4\). This tells us that for any input value \(x\), we substitute it into the expression \(-3x + 4\) to find the output.

For instance, if we want to evaluate \(f(2)\), we substitute 2 into \( x\) and calculate:
\(f(2) = -3(2) + 4\)

This simplifies to: \(f(2) = -6 + 4 = -2\).

So, evaluating functions is essentially about plugging in the input value and simplifying to find the result.

Substitution is a key technique when working with functions. It involves replacing a variable with a specific value, expression, or another variable. Let's see it in action with our function example.

We're given \(f(x) = -3x + 4\) and asked to evaluate \(f(2t + 1)\). This means we need to substitute \(x\) with \(2t + 1\).

So, we rewrite the function as: \(f(2t + 1) = -3(2t + 1) + 4\).

This is what substitution looks like—replacing the \(x\) variable with the expression \(2t + 1\). Now, we need to simplify the new expression.

Simplifying expressions is the process of making a mathematical expression as simple as possible. It usually involves combining like terms and performing arithmetic operations.

Starting from our substituted function \(f(2t + 1) = -3(2t + 1) + 4\), the first step is to distribute the \(-3\) across the terms inside the parentheses:

\[f(2t + 1) = -3 \times 2t + -3 \times 1 + 4\]

This becomes:
\[f(2t + 1) = -6t - 3 + 4\].

Next, combine the constant terms \(-3\) and \(+4\):
\[-6t - 3 + 4 = -6t + 1\].

So, the simplified form of our expression \(f(2t + 1)\) is \(-6t + 1\).

Remember, simplifying involves arithmetic operations like addition, subtraction, and sometimes distribution, to make an expression as concise and clear as possible.