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

검증된 단계별 안내

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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주요 개념

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Function Operations (Subtraction)

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.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. When performing operations like subtraction on functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all expressions are valid.

Polynomial Functions and Their Domains

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.