Questions: Given that f(x)=x^2-9x and g(x)=x+15, find (a) (f+g)(x)= (b) (f-g)(x)= (c) (f g)(x)= (d) (f/g)(x)=

Solution Steps

To solve the given problems, we need to perform basic operations on the functions \( f(x) \) and \( g(x) \). Specifically, we will: (a) Add the functions \( f(x) \) and \( g(x) \). (b) Subtract \( g(x) \) from \( f(x) \). (c) Multiply the functions \( f(x) \) and \( g(x) \).

To find \( (f + g)(x) \), we add the two functions: \[ f(x) = x^2 - 9x \] \[ g(x) = x + 15 \] Thus, \[ (f + g)(x) = f(x) + g(x) = (x^2 - 9x) + (x + 15) = x^2 - 8x + 15 \] For \( x = 5 \): \[ (f + g)(5) = 5^2 - 8 \cdot 5 + 15 = 25 - 40 + 15 = 0 \]

To find \( (f - g)(x) \), we subtract \( g(x) \) from \( f(x) \): \[ (f - g)(x) = f(x) - g(x) = (x^2 - 9x) - (x + 15) = x^2 - 10x - 15 \] For \( x = 5 \): \[ (f - g)(5) = 5^2 - 10 \cdot 5 - 15 = 25 - 50 - 15 = -40 \]

To find \( (f \cdot g)(x) \), we multiply the two functions: \[ (f \cdot g)(x) = f(x) \cdot g(x) = (x^2 - 9x)(x + 15) \] Expanding this: \[ = x^3 + 15x^2 - 9x^2 - 135x = x^3 + 6x^2 - 135x \] For \( x = 5 \): \[ (f \cdot g)(5) = 5^3 + 6 \cdot 5^2 - 135 \cdot 5 = 125 + 150 - 675 = -400 \]

Final Answer