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Function notation is a way of representing a function in mathematics. It typically uses a letter, like \( f \), followed by \( (x) \), which signifies that \( x \) is the input variable. This notation is a handy way to express equations where each input has a corresponding output, almost like a machine that turns inputs into outputs.
For example, in the function \( f(x) = -5x + 1 \), \( f(x) \) indicates a function named \( f \) with \( x \) as the variable. The expression \(-5x + 1\) shows how the output is calculated from \( x \).
When we say \( f(x) = 2 \), we're interested in finding out what \( x \) will produce an output of 2 when plugged into the function. This represents finding a specific input, where the machine outputs 2.

Solving for \( x \) means finding the value of \( x \) that makes an equation true. Let's walk through the process:
- **Identify the equation:** For this exercise, we start with \(-5x + 1 = 2\). This just means we need to find what \( x \) should be so that \(-5x + 1\) equals 2.- **Use inverse operations:** We want to isolate \( x \). So, first, get rid of any additional numbers or coefficients on the same side as \( x \). Here, you'd subtract 1 from both sides to cancel out the \(+ 1\).
- **Perform operations step by step:** Follow through by solving step-by-step—subtract first, then divide. Here, once \(-5x = 1\), divide by the coefficient of \( -5 \) to solve for \( x \).The result is \( x = -\frac{1}{5} \), which aligns with our objective of finding the value that satisfies the equation.

Algebraic manipulation involves using basic operations to transform an equation into a simpler form where the unknown can be easily found. It's like cleaning a messy room until you find what you're looking for.
Here's how it unfolds in our exercise:
- **Start with substitution:** In this context, we start by substituting \( f(x) = 2 \) into the function equation, leading to the equation \(-5x + 1 = 2\).- **Rearranging terms:** The goal is to separate terms with \( x \) from the constants. Subtract 1 on both sides to focus on the term with \( x \).- **Solving isolation:** Algebraic manipulation is not only about moving things around—it's about finding ways to simplify. Dividing both sides by \(-5\) lets us isolate \( x \) on one side.
This step-by-step isolation and simplification lead us to find that \( x = -\frac{1}{5} \). Mastering this skill can make solving equations much more straightforward.