Problem 43 - Algebra 2

Question

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(f\left(\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

\( f\left(\frac{1}{2}\right) = 5 \)

Step 1: Understand the Function

The function given is \( f(x) = 6x + 2 \). To find \( f\left(\frac{1}{2}\right) \), substitute \( x = \frac{1}{2} \) into the function.

Step 2: Substitute the Value

Replace \( x \) with \( \frac{1}{2} \) in \( f(x) \). This gives us:\[ f\left(\frac{1}{2}\right) = 6 \left( \frac{1}{2} \right) + 2 \]

Step 3: Simplify the Expression

Simplify \( 6 \left( \frac{1}{2} \right) \) first, which equals \( 3 \). So we have:\[ 3 + 2 \]

Step 4: Calculate the Result

Add the numbers together to find the final result:\[ 3 + 2 = 5 \]

Key Concepts

Substitute Value into Function

When asked to find a specific output of a function, the first step is to substitute a given value into the function. This step involves replacing the variable in the function equation with the given number. In the example of the function \( f(x) = 6x + 2 \), where we need to find \( f\left(\frac{1}{2}\right) \), we replace every instance of \( x \) with \( \frac{1}{2} \).

• Identify the variable in the function equation.
• Substitute the given value of the variable into the equation.
• Make sure to correctly replace all instances of the variable.

This substitution changes the function from a general rule applicable to any value, to a specific calculation for a given value. After substituting, the function looks like \( f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right) + 2 \). By substituting correctly, you set the stage for simplifying the expression.

Simplify Expression

Once the value is substituted into the function, as with \( f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right) + 2 \), the next essential step is to simplify the expression. Simplification entails performing any arithmetic operations within the function, following the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

• Perform multiplication, division, addition, and subtraction as needed.
• Always simplify step by step to avoid mistakes.

In the given example, the multiplication \( 6 \times \frac{1}{2} \) simplifies to 3. After this, the expression becomes \( 3 + 2 \). This structured simplification approach ensures clarity and accuracy in reaching the final simplified expression.

Linear Function Calculation

Linear functions, like \( f(x) = 6x + 2 \), represent straight lines when graphed on a coordinate plane. These functions follow a simple arithmetic rule: for any input \( x \), multiply it by the coefficient (6 in this case), then add the constant term (2 here). Calculating the outcome involves a straightforward sequence of operations.

• Multiply the input value by the function's coefficient.
• Add the constant term to this product.
• This sum provides the output of the function for that specific input.

In our example, after substituting and simplifying, we get \( 3 + 2 \) which gives a final result of 5. This simple pattern in linear functions allows for quick calculations and a clear understanding of how different inputs affect the output.