Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{I}(\mathrm{f}(2)) $$

Step-by-Step Solution

Answer

The function value \( f(2) \) is 8.

Step 1: Understand the Given Function

The problem gives us the linear function \( f(x) = 4x \). This means for any value of \( x \), the output is obtained by multiplying \( x \) by 4.

Step 2: Substitute the Given Value

We need to find \( f(2) \). To do this, substitute \( x = 2 \) into the function \( f(x) = 4x \).

Step 3: Perform the Calculation

Substitute \( x = 2 \) into the function: \[ f(2) = 4 \times 2 \].

Step 4: Simplify the Expression

Simplify the expression obtained in Step 3: \[ f(2) = 8 \]. Thus, the function value when \( x = 2 \) is 8.

Key Concepts

Function Evaluation

Evaluating a function involves finding the output for a given input based on a specific rule. In the context of a linear function, like \( f(x) = 4x \), the evaluation process allows us to understand how the function behaves with different values of \( x \).
To evaluate a function, follow these simple steps:

• Identify the function rule, which in this case is multiplying the input \( x \) by 4.
• Determine the input value you're asked to evaluate, such as \( f(2) \).
• Apply the rule to the input value to find the output.

Function evaluation is a fundamental skill in algebra that helps us explore relationships between variables and predict outcomes.

Substitution Method

The substitution method is a straightforward technique used to solve equations and evaluate functions by replacing a variable with a given value. In the problem at hand, we need to evaluate \( f(x) = 4x \) at \( x = 2 \).
To use the substitution method, follow these steps:

• Identify the expression where the substitution will occur—in this case, \( 4x \).
• Replace every instance of \( x \) with the given value, which is 2 here. So the expression becomes \( 4 \times 2 \).
• After substitution, carry out the arithmetic operations to find the result.

The substitution method is a key tool in algebra, enabling the evaluation of expressions under specific conditions.

Simplifying Expressions

Simplifying expressions is the process of making an expression as straightforward as possible by performing all possible arithmetic operations. When we evaluate a function, simplification is often the final step to uncover the simplest form of the answer.
Here’s how simplification works in our function example:

• Once we substitute \( x = 2 \) into the function, we have \( 4 \times 2 \).
• Perform the multiplication to get 8.
• Since there are no more arithmetic operations left, 8 is the simplified value of the expression \( f(2) \).

Simplifying helps in clear communication of mathematical ideas, ensuring that answers are easy to read and understand.