91Ó°ÊÓ

Substitution is a fundamental concept in mathematics. It involves replacing a variable in an expression or function with a specific value. This technique helps to evaluate or simplify functions and equations. Let's break it down with our example. Given the functions,

• \( f(x) = -2x + 3 \)
• \( g(x) = x^2 - 5 \)

we need to find the value of these functions at \( x = -1 \).
For \( f(x) \), we substitute \( -1 \) for \( x \):

\( f(-1) = -2(-1) + 3 \)

Solving this gives \( f(-1) = 5 \). Similarly, for \( g(x) \), we substitute \( -1 \) for \( x \):

\( g(-1) = (-1)^2 - 5 \)

Solving this gives \( g(-1) = -4 \).
Substitution is an essential skill that simplifies solving equations and understanding function behavior.

Function operations are mathematical procedures applied to functions. This can include addition, subtraction, multiplication, and division of functions. Understanding these operations allows us to combine and manipulate functions effectively. In our problem, we have two functions:

• \( f(x) = -2x + 3 \)
• \( g(x) = x^2 - 5 \)

After evaluating these functions at \( x = -1 \), we obtained \( f(-1) = 5 \) and \( g(-1) = -4 \). Now, we need to consider the operation of multiplication applied to these evaluated results. This leads us to:

\( f(-1) \times g(-1) = 5 \times -4 = -20 \)

Understanding function operations helps in solving more complex problems involving multiple functions.

Multiplication is one of the basic arithmetic operations that you apply when combining values. In the realm of functions, we can multiply two functions after evaluating them at a common point. Let's revisit the result from our previous steps where we found:

• \( f(-1) = 5 \)
• \( g(-1) = -4 \)

To find \( f(-1) \times g(-1) \), we multiply the results together:

\( f(-1) \times g(-1) = 5 \times -4 = -20 \)

This final result shows how multiplication of evaluated function values works. It's a straightforward yet powerful operation that is commonly used in various mathematical problems.