91Ó°ÊÓ

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. It helps in expressing mathematical relationships, handling unknown values, and solving equations. In the context of functions, algebraic manipulation allows us to combine and evaluate different functions.
When working with algebra in functions, it's essential to understand a few key points:

• Variables: Symbols like \(x\) and \(y\) that can take different values.
• Expressions: Combinations of variables and constants using operations like addition and multiplication.
• Equations: Mathematical statements indicating that two expressions are equal.

By understanding these core components, it becomes easier to work with function operations and evaluations.

Evaluating functions involves finding the value of a function for a specific input value. Here's how you can evaluate a function step-by-step:
1. Identify the function you need to evaluate. For example, let's consider \(h(x) = x - 3\).
2. Substitute the given input value into the function. If we need to evaluate \(h(-2)\), we replace \(x\) with \(-2\).
3. Perform the calculations. In this case: \[ h(-2) = -2 - 3 = -5 \]
So, \( h(-2) = -5 \).
This method applies to any function, whether it's linear, quadratic, or more complex. By following these steps, you can systematically find the output for any given input.

Function operations refer to ways you can combine different functions, such as addition, subtraction, multiplication, and composition.
In the context of the given problem, we focus on function composition. Function composition means applying one function to the results of another function. Here's a step-by-step guide:
1. Identify the functions: \( g(x) = 2x \) and \( h(x) = x - 3 \).
2. Find the composite function, in this case, \((gh)(x)\), which means applying \(h(x)\) first and then \(g(x)\).
3. Evaluate \(h(x)\) for the given input. For \( x = -2 \): \[ h(-2) = -2 - 3 = -5 \]
4. Use the result of \( h(x) \) as the input for \( g(x) \). So, \( g(h(-2)) = g(-5) \): \[ g(-5) = 2 \times -5 = -10 \]
So, in this example, the result of\((gh)(-2) \) is \(-10\).
By understanding these operations, we can handle more complex expressions and solve various mathematical problems involving multiple functions.