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To add two functions together, you simply sum up their values for any input. Let's break it down: if you have two functions, say \(f(x)\) and \(g(x)\), then the new function \((f + g)(x)\) is created by adding \(f(x)\) and \(g(x)\). In our example:

• \(f(x) = x^2 + 3\)
• \(g(x) = -2x + 6\)

To find \((f + g)(x)\), we compute \(f(x) + g(x)\). When you need the value of this new function at a specific point, say \(x = -5\), you substitute \(-5\) into both \(f(x)\) and \(g(x)\). Then, add the results. Here’s a step-by-step:
1. Calculate \(f(-5)\).
2. Calculate \(g(-5)\).
3. Add the results of \(f(-5)\) and \(g(-5)\) to get \((f + g)(-5)\).
Function addition simplifies complicated problems by breaking them into smaller parts.

Substitution is a core skill in math, especially when dealing with functions. It means replacing the variable in a function with a specific value. Let’s use \(f(x) = x^2 + 3\) as an example. To find \(f(-5)\), we substitute \(-5\) for \(x\) in the function.
Here’s how you do it:

• Replace every occurrence of \(x\) with \(-5\).
• So, \(f(-5) = (-5)^2 + 3\).

This becomes: \(f(-5) = 25 + 3 = 28\).
Likewise, we find \(g(-5)\) for the function \(g(x) = -2x + 6\) by substituting \(-5\) for \(x\).

• Replace \(x\) with \(-5\).
• So, \(g(-5) = -2(-5) + 6\).

This solves to: \(g(-5) = 10 + 6 = 16\).
Substitution allows us to find the specific values of functions at given points easily.

Evaluating functions means finding their value at specific points. In our case, we evaluate both \(f(x)\) and \(g(x)\) at \(x = -5\). Here’s what we did:

• First, calculate \(f(-5)\): Substitute \(-5\) into \(f(x) = x^2 + 3\) and solve. We get \(f(-5) = 28\).
• Next, calculate \(g(-5)\): Substitute \(-5\) into \(g(x) = -2x + 6\) and solve. We get \(g(-5) = 16\).

After evaluating these functions individually, we then added them together: \((f + g)(-5) = f(-5) + g(-5)\).
Thus, \((f + g)(-5) = 28 + 16 = 44\).
Evaluating functions step-by-step ensures we get accurate results. Each step should be carefully handled to avoid errors.