Problem 57 - Algebra 2

Question

CHALLENGE If \(f(3 a-1)=12 a-7,\) find one possible expression for \(f(x)\)

Step-by-Step Solution

Answer

One possible expression for \(f(x)\) is \(f(x) = 4x - 3\).

Step 1: Understand the Problem

The function given can be expressed as a relationship between its input, which is in the form \(3a - 1\), and its output, \(12a - 7\). Our goal is to express this function in terms of \(x\), where \(x = 3a - 1\).

Step 2: Express x in terms of a

Identify \(x\) as the input in \(3a - 1\). Set \(x = 3a - 1\) and solve for \(a\):\[a = \frac{x + 1}{3}.\]

Step 3: Substitute a in the output expression

Now, substitute \(a = \frac{x + 1}{3}\) into the output expression \(12a - 7\):\[ f(x) = 12\left(\frac{x + 1}{3}\right) - 7.\]

Step 4: Simplify the Expression

Simplify the expression for \(f(x)\): \[ f(x) = 4(x + 1) - 7 = 4x + 4 - 7 = 4x - 3.\]

Step 5: Verify the Function

To verify, substitute \(x = 3a - 1\) back into \(f(x) = 4x - 3\) and simplify to check if it equals \(12a - 7\):\[ f(3a - 1) = 4(3a - 1) - 3 = 12a - 4 - 3 = 12a - 7.\] The function is verified.

Key Concepts

Function Transformation

Function transformation involves altering the way a function looks or behaves while maintaining the essential relationships between its variables. When we encounter something like transforming the function from terms of a variable \(a\) to \(x\), we are performing a specific type of function transformation. This is vital when you need to compare or analyze the function under different conditions or variables.

In this problem, we started with a function given in terms of \(3a - 1\) and transformed it into a function where the input is \(x\). This transformation involves a couple of steps:

• Pretend you are rebranding the function by substituting the input \(3a - 1\) with \(x\).
• Then, rewrite or manipulate the function to show how it works with \(x\) as the input instead of \(3a - 1\).

This transformation aids in simplifying the function and understanding its behavior, allowing us to explore the function's potential with different inputs. By completing these steps, we were able to clearly see how the original function relates to its new form using \(x\). This kind of transformational insight is key in many areas of mathematics.

Input-Output Relationship

A core concept in understanding functions is the input-output relationship. This idea centers around how the input, or what you "feed" into a function, relates to the output, or what you "get out" of the function. In mathematical terms, if you put \(x\) into the function \(f\), you get \(f(x)\).

For the given exercise, the input-output relationship is clearly demonstrated. Initially, with the input \(3a - 1\), the output is \(12a - 7\). By transforming the function to use \(x\), the new input-output relationship becomes \(x\) to \(f(x) = 4x - 3\).

This relationship tells us:

• If we change our input, our output will also change accordingly based on the rule or equation defining the function.
• It's consistent, meaning every time you put the same input in, you'll get the same output.

Understanding this relationship not only helps in solving problems but also develops a deeper appreciation for how functions model real-world scenarios.

Algebraic Manipulation

Algebraic manipulation refers to various techniques used to rearrange and simplify expressions and equations. In our exercise, we utilized algebraic manipulation to transition from \(f(3a - 1) = 12a - 7\) to express \(f(x)\) directly.

Let's break down the manipulation process involved:

• First, convert the known input \(3a - 1\) into a variable \(x\) to simplify the problem.
• Next, solve for \(a\) in terms of \(x\) by rearranging the equation: \(a = \frac{x + 1}{3}\).
• Substitute this into the function expression for \(12a - 7\), rephrasing the function in terms of \(x\): \(12 \left(\frac{x + 1}{3}\right) - 7\).
• Simplify to find the neat expression of the function \(f(x) = 4x - 3\).

These steps highlight how algebraic techniques are powerful tools for making expressions easier to work with. Such manipulations are crucial for both simplifying problems and for providing clearer insights into the relationships within mathematical models.