91Ó°ÊÓ

To find the addition of the two functions, f(x) + g(x), we need to add the two functions: \(f(x) + g(x) = (3x) + (x^2 - 1)\) Now simplify the expression: \(f(x) + g(x) = x^2 + 3x - 1\)

To find the subtraction of the two functions, f(x) - g(x), we need to subtract g(x) from f(x): \(f(x) - g(x) = (3x) - (x^2 - 1)\) Now simplify the expression: \(f(x) - g(x) = -x^2 + 3x + 1\)

To find the multiplication of the two functions, f(x) * g(x), we need to multiply the two functions: \(f(x) * g(x) = (3x) * (x^2 - 1)\) Now expand and simplify the expression: \(f(x) * g(x) = 3x^3 - 3x\)

To find the division of the two functions, f(x) / g(x), we need to divide f(x) by g(x): \(f(x) / g(x) = \frac{3x}{x^2 - 1}\)

Now, we need to find the domain of each resulting function. (a) f(x) + g(x) = x^2 + 3x - 1 The domain for this function is all real numbers since there are no restrictions. Domain: \((-\infty, \infty)\) (b) f(x) - g(x) = -x^2 + 3x + 1 The domain for this function is also all real numbers since there are no restrictions. Domain: \((-\infty, \infty)\) (c) f(x) * g(x) = 3x^3 - 3x The domain for this function is all real numbers since there are no restrictions. Domain: \((-\infty, \infty)\) (d) f(x) / g(x) = \(\frac{3x}{x^2 - 1}\) For this function, we need to make sure that the denominator is not equal to zero, as division by zero is not allowed. So, the domain will include all real numbers except for those values that make the denominator zero. Set the denominator equal to zero to find the excluded values: \(x^2 - 1 = 0\) Solving for x, we get \(x = \pm 1\) So, the domain of this function is all real numbers except ±1. Domain: \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)