Problem 105 - Algebra A Combined Function

Question

For each function, find the indicated values. \(f(x)=x-12\); a. \(f(12)\) b. \(f(a)\) c. \(f(-x)\) d. \(f(x+h)\)

Step-by-Step Solution

Answer

a. 0; b. \(a - 12\); c. \(-x - 12\); d. \(x + h - 12\).

Step 1: Substitute and Evaluate f(12)

To find \(f(12)\), substitute 12 for \(x\) in the function. This gives: \[f(12) = 12 - 12.\] Calculate the result: \[f(12) = 0.\]

Step 2: Evaluate f(a) with Substitution

To find \(f(a)\), substitute \(a\) for \(x\) in the function. Thus, the function becomes: \[f(a) = a - 12.\] This expression is the value of \(f(a)\).

Step 3: Substitute for f(-x) and Simplify

To find \(f(-x)\), substitute \(-x\) for \(x\) in the function. This results in: \[f(-x) = -x - 12.\] The function simplifies to \(-x - 12\).

Step 4: Evaluate f(x + h) by Substitution

For \(f(x + h)\), substitute \(x + h\) for \(x\) in the original function: \[f(x + h) = (x + h) - 12.\] Simplifying gives: \[f(x + h) = x + h - 12.\]

Key Concepts

Substitution

Substitution is a key concept in mathematics, especially when working with functions. It involves replacing a variable in an expression or equation with a specific value or another expression. This allows us to evaluate or transform functions in specific ways.

• For example, in the function \(f(x) = x - 12\), if we substitute \(x\) with 12, we get \(f(12) = 12 - 12 = 0\).
• Substitution lets you see how the function behaves with different inputs.
• It's crucial for solving equations and sometimes for simplifying expressions.

By using substitution, you replace variables with numbers or other variables to better understand function outcomes. Remember, it is a tool to "test" functions under specific conditions, which can be very informative.

Simplifying Expressions

Simplifying expressions is another essential process when dealing with functions and algebra in general. Once you substitute variables in equations, the expression may need simplification.

• Simplification involves combining like terms and performing operations to reduce the expression to its simplest form.
• For instance, when we find \(f(-x) = -x - 12\), the expression is already simplified.
• If you have an expression like \(f(x+h) = x + h - 12\), ensure all terms that can be combined are merged, and any unnecessary complexity is removed.

This process requires understanding how to handle algebraic expressions systematically. The goal is to make expressions as concise as possible, making them easier to understand and solve.

Algebraic Functions

Algebraic functions are mathematical expressions involving variables, constants, and the operations of addition, subtraction, multiplication, and division. These functions are central to algebra and calculus.

• In our example, \(f(x) = x - 12\) is a simple linear algebraic function.
• Algebraic functions can be used to model real-world situations, providing insights into how changing one variable affects the overall system.
• Understanding how to evaluate and manipulate these functions is vital for solving equations and graphing.

They form the building blocks of many advanced mathematical theories. Grasping their properties through substitution and simplification can greatly enhance your problem-solving skills.