91Ó°ÊÓ

In the context of evaluating functions, substitution is a crucial step. It involves replacing a variable with a given expression or number. For example, consider the function \( f(x) = -3x + 4 \). If you need to evaluate \( f(3t - 2) \), you substitute \(3t - 2\) in place of \(x\) in the function.
Here’s how you do it:
- Start with the function equation: \( f(x) = -3x + 4 \)
- Substitute the expression \(3t - 2\) for \(x\): \[ f(3t - 2) = -3(3t - 2) + 4 \]
Substitution transforms the function into a new form, making it possible to simplify and find the value for different situations.

Function composition involves applying one function to the results of another function. Although our problem didn't involve multiple functions, understanding this concept is essential.
Think of two functions, \( f(x) \) and \( g(x) \). Function composition is noted as \( (f \, ∘ \, g)(x) \), which means \( f(g(x)) \). For example, if \( f(x) = -3x + 4 \) and \( g(x) = -x^2 + 4x + 1 \), then \( f(g(x)) \) means you first apply \( g(x) \) and then use the result as input to \( f \).
This concept is useful for solving more complex problems where multiple functions interact. It allows you to layer different functional operations and find the composite value.

Simplification is the process of reducing an expression into its simplest form. This involves performing arithmetic operations and combining like terms.
Let’s revisit our example with \( f(x) = -3x + 4 \) and we need to find \( f(3t - 2) \). Here’s a step-by-step simplification:
- After substituting \(3t - 2\) for \(x\), we had: \[ f(3t - 2) = -3(3t - 2) + 4 \]
- Distribute \(-3\): \[-3(3t) + (-3)(-2) = -9t + 6 \]
- Combine like terms with the constant: \[ f(3t - 2) = -9t + 6 + 4 \]
- Simplify further by adding the constants: \[ f(3t - 2) = -9t + 10 \]
Simplification makes the expression easier to understand and evaluate, showing the final, most reduced form. Remember, always perform operations systematically to avoid mistakes.