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Function subtraction is a way to find the difference between two functions. If we have two functions, say, \(f(x)\) and \(g(x)\), the difference is represented as \((f - g)(x)\). This means we subtract the value of \(g(x)\) from \(f(x)\) for the same \(x\). The formula looks like this: \[ (f - g)(x) = f(x) - g(x) \] In our example: \[ f(x) = x^2 + 3 \] and \[ g(x) = -2x + 6 \]. To get the function \(f - g\) we need to subtract \(g(x)\) from \(f(x)\). So, for any value of \(x\), \[ (f - g)(x) = (x^2 + 3) - (-2x + 6) \]. Subtracting those components individually, we get: \[ x^2 + 3 + 2x - 6 \]. This simplifies to: \[ (f - g)(x) = x^2 + 2x - 3 \]. Function subtraction is crucial when analyzing the differences between functions' outputs at specific inputs.

Finding function values at specific points are about evaluating the function by substituting the given value of \(x\) into the formula. Let's take the functions \[ f(x) = x^2 + 3 \] and \[ g(x) = -2x + 6 \]. If we need to determine \(f(-1)\), we'll substitute \(-1\) into \(x\) in the function \(f(x)\): \[ f(-1) = (-1)^2 + 3 \]. Calculating the expression inside: \[ (-1)^2 \] gives 1. So: \[ 1 + 3 = 4 \]. Likewise, to find \(g(-1)\), we'll substitute \(-1\) into \(x\) in the function \(g(x)\): \[ g(-1) = -2(-1) + 6 \]. Solving the expression: \[ 2 + 6 = 8 \]. Thus, \[ f(-1) = 4 \] and \[ g(-1) = 8 \]. Evaluating functions at specific values provides outputs needed to perform further operations or comparisons.

The substitution method involves replacing a variable with a specific value to solve or simplify an expression. In our given example, we use substitution to find function values. Let's see how: For the functions \[ f(x) = x^2 + 3 \] and \[ g(x) = -2x + 6 \], we substitute \(x = -1\) to find \(f(-1)\) and \(g(-1)\). First, replacing \(x\) in \(f\):\[ f(-1) = (-1)^2 + 3 \]. Calculate: \[ f(-1) = 1 + 3 = 4 \]. Next, replacing \(x\) in \(g\):\[ g(-1) = -2(-1) + 6 \]. Calculate: \[ g(-1) = 2 + 6 = 8 \]. Substitution is a powerful technique in mathematics for solving equations and evaluating expressions. It simplifies complex expressions and helps in understanding behavior of functions at specific points.