Problem 60 - Algebra 2

Question

Find each value if \(f(x)=3 x-4\). $$ f(-1) $$

Step-by-Step Solution

Answer

The value of \( f(-1) \) is \(-7\).

Step 1: Substitute the Input into the Function

To find \( f(-1) \), substitute \( -1 \) for \( x \) in the function \( f(x) = 3x - 4 \). This gives us the expression \( f(-1) = 3(-1) - 4 \).

Step 2: Simplify the Expression

Simplify the expression by first multiplying: \( 3(-1) = -3 \). Then continue simplifying: \( -3 - 4 = -7 \).

Key Concepts

Function Evaluation

When dealing with linear functions like \(f(x) = 3x - 4\), function evaluation is an essential skill. Essentially, it involves finding out the value of a function given a specific input. In the exercise provided, the input is \(-1\) and we need to determine what \(f(-1)\) equals.

• First, identify the function. Here it's \(f(x) = 3x - 4\).
• Next, recognize the given input. Here it's \(-1\).
• The goal is to substitute this value into the function in place of \(x\).

This allows us to "evaluate" or find the specific output for the given input. Function evaluation helps in understanding how different inputs affect the output of the function.

Substitution

Substitution is the step where we plug in our specific input value into the given function. In this case, we substitute \(-1\) into the function \(f(x) = 3x - 4\). This is how it is done:

• Take the function: \(f(x) = 3x - 4\).
• Replace \(x\) with \(-1\) to get: \(f(-1) = 3(-1) - 4\).

It's like replacing one piece of a puzzle with another. By doing this, we transform a generalized expression into a specific one that can be calculated. Substitution is a vital algebraic process that helps in pinpointing exact values from abstract expressions.

Algebraic Simplification

Once substitution is completed, the next step is algebraic simplification. This process involves simplifying the expression we've substituted to find the final answer. For our current example, the expression became \(f(-1) = 3(-1) - 4\).

• First, perform the multiplication: \(3(-1) = -3\).
• Then perform the subtraction: \(-3 - 4 = -7\).

Each operation should be handled step by step, ensuring accuracy.
Simplification turns complex expressions into simple, easily digestible numbers or terms. It's crucial in solving algebraic problems and is often the final step in evaluating functions.