Given that \( f(x)=x^{2}-4 x \) and \( g(x)=x+4 \), find (a) \( (f+g)(x)= \) (b) \( (f-g)(x)= \) प (c) \( (f g)(x)= \) (d) \( \left(\frac{f}{g}\right)(x)= \) Question Help:

To solve the problem, we will find the sum, difference, product, and quotient of the functions \( f(x) \) and \( g(x) \). Given: - \( f(x) = x^2 - 4x \) - \( g(x) = x + 4 \) Let's solve each part step by step. ### (a) \( (f+g)(x) \) The sum of the functions \( f \) and \( g \) is given by: \[ (f+g)(x) = f(x) + g(x) \] Substituting the expressions for \( f(x) \) and \( g(x) \): \[ (f+g)(x) = (x^2 - 4x) + (x + 4) \] Now, combine like terms: \[ (f+g)(x) = x^2 - 4x + x + 4 = x^2 - 3x + 4 \] ### (b) \( (f-g)(x) \) The difference of the functions \( f \) and \( g \) is given by: \[ (f-g)(x) = f(x) - g(x) \] Substituting the expressions for \( f(x) \) and \( g(x) \): \[ (f-g)(x) = (x^2 - 4x) - (x + 4) \] Now, distribute the negative sign and combine like terms: \[ (f-g)(x) = x^2 - 4x - x - 4 = x^2 - 5x - 4 \] ### (c) \( (fg)(x) \) The product of the functions \( f \) and \( g \) is given by: \[ (fg)(x) = f(x) \cdot g(x) \] Substituting the expressions for \( f(x) \) and \( g(x) \): \[ (fg)(x) = (x^2 - 4x)(x + 4) \] Now, we will expand this expression: \[ (fg)(x) = x^2 \cdot (x + 4) - 4x \cdot (x + 4) \] \[ = x^3 + 4x^2 - 4x^2 - 16x \] \[ = x^3 - 16x \] ### (d) \( \left(\frac{f}{g}\right)(x) \) The quotient of the functions \( f \) and \( g \) is given by: \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \] Substituting the expressions for \( f(x) \) and \( g(x) \): \[ \left(\frac{f}{g}\right)(x) = \frac{x^2 - 4x}{x + 4} \] This expression can be simplified if possible. We can factor the numerator: \[ = \frac{x(x - 4)}{x + 4} \] Now, we have all the required expressions: ### Final Answers: (a) \( (f+g)(x) = x^2 - 3x + 4 \) (b) \( (f-g)(x) = x^2 - 5x - 4 \) (c) \( (fg)(x) = x^3 - 16x \) (d) \( \left(\frac{f}{g}\right)(x) = \frac{x(x - 4)}{x + 4} \)