Find f−g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

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Identify the given functions: $f(x) = x - 5$ and $g(x) = 3x^{2}$.

Find the difference of the functions, which is $f - g$. This means subtract $g(x)$ from $f(x)$: write the expression as $f(x) - g(x)$.

Substitute the given functions into the expression: $f(x) - g(x) = (x - 5) - (3x^{2})$.

Simplify the expression by removing parentheses and combining like terms: $x - 5 - 3x^{2}$.

Determine the domain of the new function $f - g$. Since both $f(x)$ and $g(x)$ are polynomials, their domain is all real numbers, so the domain of $f - g$ is also all real numbers.

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Function subtraction involves creating a new function by subtracting the output of one function from another for the same input value. For functions f and g, (f - g)(x) = f(x) - g(x). This operation combines the two functions algebraically and is essential for manipulating and analyzing function behavior.

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Polynomial functions, such as f(x) = x - 5 and g(x) = 3x², are defined for all real numbers because they involve only powers of x with real coefficients. Understanding that polynomials have domains of all real numbers simplifies determining the domain of combined functions like f - g.

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