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To begin solving any function-related problem, it's crucial to understand what function evaluation means. At its core, function evaluation involves determining a specific output for an input value in a given function. A function can be thought of as a machine that takes an input, processes it, and gives an output. When you "evaluate a function," you're essentially asking what output you get when you put a particular input into the function.
A function is typically written as \( f(x) \), where \( x \) is the input variable. For the function \( f(x) = (x+5)^2 \), we evaluate it by substituting different values of \( x \) and calculating the result. In our example, we're asked to find values of \( x \) such that \( f(x) = 3 \). This means we're searching for values that, when inserted into the function, will yield an output of 3.
By setting \( f(x) = 3 \), you obtain the equation \( (x+5)^2 = 3 \), setting the stage for further solving. Understanding this setup is critical in the process of solving any function-related question.

Once the function is set as an equation, like \( (x+5)^2 = 3 \), the next step is to solve it. A common method used here is the square root method. This involves taking the square root of both sides of the equation and is a useful method when dealing with quadratic equations like ours.
Imagine the equation \( y^2 = c \). In such situations, resolving \( y \) involves taking the square root of \( c \). Be mindful; square roots have both positive and negative values. That means \( y = \sqrt{c} \) and \( y = -\sqrt{c} \) are both solutions.
Applying this to our equation \( (x+5)^2 = 3 \) gives us two scenarios: \( x+5 = \sqrt{3} \) and \( x+5 = -\sqrt{3} \). This step is important as it effectively breaks down the quadratic equation into linear parts, simplifying the solving process.

With the square root taken on both sides, our task becomes isolating \( x \). Isolating refers to arranging the equation such that \( x \) stands by itself on one side of the equation. This is often the primary goal in solving an equation.
In our problem structure, from \( x+5 = \sqrt{3} \) and \( x+5 = -\sqrt{3} \), isolating \( x \) involves subtracting 5 from both sides. Thus:

• For the equation \( x+5 = \sqrt{3} \), we subtract 5 giving us \( x = -5 + \sqrt{3} \).
• And for \( x+5 = -\sqrt{3} \), it simplifies to \( x = -5 - \sqrt{3} \).

These steps allow you to unravel the final solutions effectively, providing us with the values of \( x \) that satisfy the original function equation. Always remember, isolating \( x \) is about balancing the equation to express \( x \) clearly and correctly.