91Ó°ÊÓ

Substitution is a fundamental concept in algebra and calculus. It's the process of replacing a variable in an expression with another value or expression. In our exercise, we started with the function definition:

• Given function: f(x) = -3x + 4

We need to find f(x-2). This requires us to substitute the variable x in the function with (x-2). This step changes the function to:

• Substitution: f(x-2) = -3(x-2) + 4

By substituting (x-2), we transform the original function, helping us evaluate it at a new input. This method is essential for solving a wide range of mathematical problems, especially when dealing with complex functions and equations.
Understanding substitution can also help in calculus, where we often change variables to simplify integration or differentiation processes..

Algebraic manipulation involves rearranging and simplifying expressions and equations. It's a critical skill for solving mathematical problems. In our solution, we used algebraic manipulation to transform and simplify the function after substitution:

• Original substituted function: f(x-2) = -3(x-2) + 4

To simplify, we applied the distributive property. This property allows us to multiply a single term with each term inside a bracket:

• Distribute -3: -3(x-2) becomes -3x + 6

Combining these terms, we get a new expression:

• Simplify further: f(x-2) = -3x + 6 + 4
• Finally: f(x-2) = -3x + 10

Algebraic manipulation helps us clarify and solve equations, making it easier to understand and work with different mathematical problems.

Function evaluation is the process of finding the value of a function for a particular input. This is what we did when substituting (x-2) into f(x). The given function was:

• Original function: f(x) = -3x + 4

To evaluate this function at x-2, we substituted (x-2) for x:

• Substitution: f(x-2) = -3(x-2) + 4

After that, we simplified the expression to get the evaluated form of the function at the input (x-2). The evaluation process confirms the function's behavior with different inputs, enabling us to understand how changes in the input affect the output. This technique is applicable in various fields such as physics, economics, and engineering, where functions model real-world scenarios.

Simplifying expressions is a powerful tool in mathematics to make equations more manageable and easier to solve. In our exercise, after substitution, we had:

• Initial substitution: f(x-2) = -3(x-2) + 4

We distributed -3 across (x-2):

• Distributive step: -3(x-2) becomes -3x + 6

Next, we combined the like terms:

• Combining terms: -3x + 6 + 4 becomes -3x + 10

Thus, the simplified function is:

• Final simplified form: f(x-2) = -3x + 10

Simplifying expressions helps to break down complex problems, making them easier to understand and solve. It also provides a clearer view of the relationships between different components of an equation or function. Through practice, you will become more comfortable with these steps, building a strong foundation in algebra and other advanced mathematical fields.