Problem 100 - Algebra A Combined Function

Question

Suppose that \(y=f(x)\) and it is true that \(f(7)=50 .\) Determine whether each is true or false. A possible function is \(f(x)=10 x-20\).

Step-by-Step Solution

Answer

True, the function \( f(x) = 10x - 20 \) satisfies \( f(7) = 50 \).

Step 1: Understand the Given Information

We are given that the function \( f(x) \) evaluated at \( x = 7 \) is 50, that is \( f(7) = 50 \). We need to verify if the provided function \( f(x) = 10x - 20 \) satisfies this condition.

Step 2: Substitute into the Function

Substitute \( x = 7 \) into the proposed function \( f(x) = 10x - 20 \). This gives:\[f(7) = 10(7) - 20\]

Step 3: Perform Arithmetic Operations

Calculate the expression:\[f(7) = 10 \times 7 - 20 = 70 - 20 = 50\]Hence, the proposed function evaluates to 50 when \( x = 7 \).

Step 4: Conclusion About the Statement

Since the calculation shows \( f(7) = 50 \) using the function \( f(x) = 10x - 20 \), it confirms that the proposition \("A possible function is \( f(x) = 10x - 20 \)"\) is true.

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that use variables, exponents, and coefficients to form equations. These functions are fundamental in algebra as they define relationships between variables. In this context, our primary function is represented by \( f(x) = 10x - 20 \). This is a linear algebraic function because it includes only a single variable \( x \), raised to the power of one, and operation of multiplication and subtraction.

• **Variables**: These are symbols like \( x \) that can change or hold different values.
• **Exponents**: While our example doesn't use them prominently, exponents indicate repeated multiplication of a number.
• **Coefficients**: Numbers that are multiplied by variables, such as the 10 in \( 10x \).

Understanding how to identify and work with algebraic functions helps in solving many problems in algebra, such as deriving relationships and predicting outcomes from given inputs.

Step-by-Step Solution

Following a step-by-step approach simplifies complex algebra problems into manageable tasks. Let's break down how this method works using our example:

• **Identify Given Information**: We know \( f(7) = 50 \) and we need to test if \( f(x) = 10x - 20 \) fits this condition.
• **Substitution**: Start by substituting \( x = 7 \) into the function, leading to \( f(7) = 10(7) - 20 \).
• **Arithmetic Calculation**: Perform the calculation: \( 10 \times 7 - 20 = 70 - 20 \).
• **Conclusion**: Verify if the result equals the known value: \( f(7) = 50 \). This confirms the proposed function is correct.

By using a clear, structured strategy, you focus on one aspect of the problem at a time, which helps prevent errors and boosts understanding.

Arithmetic Operations

Arithmetic operations are the basic calculations used throughout mathematics. In algebra, they play a crucial role in transforming and simplifying expressions. Key operations include:

• **Addition**: Summing numbers or expression elements.
• **Subtraction**: Removing values, as seen in \( 10x - 20 \).
• **Multiplication**: Repeated addition, evident in \( 10 \times 7 \). Here, you multiply the coefficient by the variable’s value.
• **Division**: While not directly used in this problem, it breaks numbers into specified parts.

Performing arithmetic calculations accurately is essential, especially when dealing with function evaluations. This ensures that the numbers derived from substitutions, like \( f(7) = 50 \), are correct and satisfy the related conditions in algebraic expressions.