Problem 33 - Thinking Mathematically

Question

In Exercises 33-46, evaluate each function at the given value of the variable. \(f(x)=x-4\) a. \(f(8)\) b. \(f(1)\)

Step-by-Step Solution

Answer

The evaluated values are \(f(8) = 4\) and \(f(1) = -3\).

Step 1: Substituting \(x = 8\)

Substitute \(x = 8\) into the function \(f(x)\), i.e. evaluate \(f(8)\). The calculation will look like this: \(f(8) = 8 - 4\)

Step 2: Calculate \(f(8)\)

Perform the subtraction operation from step 1 to get the value of \(f(8)\), which is \(f(8) = 4\)

Step 3: Substituting \(x = 1\)

Now, substitute \(x = 1\) into the function \(f(x)\), i.e. evaluate \(f(1)\). The calculation will hence be: \(f(1) = 1 - 4\)

Step 4: Calculate \(f(1)\)

Perform the subtraction operation from step 3 to get the value of \(f(1)\), which is \(f(1) = -3\)

Key Concepts

Algebraic Functions

Understanding algebraic functions is foundational for exploring equations and their behaviors. An algebraic function, like the one you encounter with the equation f(x)=x-4, represents a relationship between one or more variables. In this function, for every input value of x, there is a corresponding output value which is obtained after performing the arithmetic operations on x.

Within the realm of these functions, we typically deal with polynomials, rationals, roots, and other algebraic expressions. They can be as simple as the given linear function or as complex as having multiple terms and variables. The power of algebraic functions is that they allow us to predict outcomes and model real-world situations mathematically. When you evaluate an algebraic function, such as f(8) or f(1), you're essentially finding out what the function 'outputs' when a specific 'input' is provided.

Substitution Method

The substitution method is a crucial tool in algebra, particularly when dealing with functions. It involves replacing a variable in an algebraic expression with its corresponding value. When you were asked to evaluate f(8) and f(1), you were applying the substitution method. By substituting x with 8 and 1, respectively, you transformed the abstract expression f(x) into concrete numbers.

Steps in the Substitution Method

• Identify the value to substitute into the function.
• Replace the variable in the function with the chosen value.
• Simplify the function, if necessary, to find the result.

The substitution method isn't just limited to evaluating functions. It's also a powerful technique for solving systems of equations, another common application in algebra.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics and consist of addition, subtraction, multiplication, division, and exponentiation. In the context of evaluating algebraic functions, we often use these operations to simplify the function after substituting the variables with their given values.

For instance, in the exercises provided, you used the operation of subtraction. After substituting x with 8 and 1, you were left with the expressions 8 - 4 and 1 - 4. By performing these simple subtractions, you found the respective function values, 4 and -3. Mastery of these arithmetic operations is essential as they are frequently used not only in algebra but in all areas of mathematics. Therefore, they form a solid foundation for any student looking to advance their mathematical knowledge.