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The domain of a function refers to the complete set of possible input values (usually represented by the variable \(x\)) for which the function is defined.
It represents every \(x\) value that can be used without making the function invalid. For the function \(f(x) = x^2 + 6\), the domain is all real numbers because a square is always defined.

However, with the function \(g(x) = \sqrt{1-x}\), the situation changes. The square root function, \(\sqrt{}\), is only real and defined when the expression inside is non-negative.

• Thus, the condition \(1-x \geq 0\) must be satisfied.
• Solving this inequality, we find \(x \leq 1\).

This means that the domain for \(g(x)\) is all real numbers \(x\) such that \(x \leq 1\).
When considering the domain of the product \((f g)(x)\), you must account for any restrictions that come from both functions.
Since \(g(x)\) limits \(x\) to \(x \leq 1\), the domain of \((f g)(x)\) is also \(x \leq 1\).

Adding two functions involves combining their outputs for a given input. If you have \(f(x)\) and \(g(x)\), then the sum \((f+g)(x)\) would be calculated as \(f(x) + g(x)\).
The problem provided uses the functions \(f(x) = x^2 + 6\) and \(g(x) = \sqrt{1-x}\).

• To add these, you do \((f+g)(x) = (x^2 + 6) + \sqrt{1-x}\).

The resulting expression represents the combined effect of applying both functions to the same \(x\).
Remember to remember the domain of combined functions. Since \(g(x)\) is defined only when \(x \leq 1\), \((f+g)(x)\) also shares this domain constraint.

Subtracting functions is similar to addition, but instead of combining their outputs, you find the difference. Given functions \(f(x)\) and \(g(x)\), the difference \((f-g)(x)\) is given by \(f(x) - g(x)\).
Let's use the same functions \(f(x) = x^2 + 6\) and \(g(x) = \sqrt{1-x}\).

• The subtraction yields \((f-g)(x) = (x^2 + 6) - \sqrt{1-x}\).

Just like addition, the domain is constrained by the function \(g(x)\), requiring \(x \leq 1\).
Always check the domains of the individual functions before adding or subtracting to avoid undefined results.

In function multiplication, outputs are multiplied for each input. If \(f(x)\) and \(g(x)\) are the functions being multiplied, the product \((fg)(x)\) equals \(f(x) \cdot g(x)\).
For the given functions \(f(x) = x^2 + 6\) and \(g(x) = \sqrt{1-x}\), their product will be:

• \((fg)(x) = (x^2 + 6) \cdot \sqrt{1-x}\)

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When computing function products, check for restrictions from each function to determine the product's domain.
Here, \(g(x)\) restricts \(x\) to \(x \leq 1\), which also applies to \((fg)(x)\).

Function multiplication can reveal new behaviors and should always be verified to align with domains.