Alright, let's solve the problem step-by-step.
Given:
[tex]\( f(x) = x^2 - 6x \)[/tex]
[tex]\( g(x) = x + 3 \)[/tex]
(a) To find [tex]\( (f+g)(x) \)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x) = (x^2 - 6x) + (x + 3)
\][/tex]
Combine like terms:
[tex]\[
= x^2 - 6x + x + 3 = x^2 - 5x + 3
\][/tex]
So, [tex]\( (f+g)(x) = x^2 - 5x + 3 \)[/tex].
(b) To find [tex]\( (f-g)(x) \)[/tex]:
[tex]\[
(f-g)(x) = f(x) - g(x) = (x^2 - 6x) - (x + 3)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
= x^2 - 6x - x - 3 = x^2 - 7x - 3
\][/tex]
So, [tex]\( (f-g)(x) = x^2 - 7x - 3 \)[/tex].
(c) To find [tex]\( (fg)(x) \)[/tex]:
[tex]\[
(fg)(x) = f(x) \cdot g(x) = (x^2 - 6x) \cdot (x + 3)
\][/tex]
Distribute [tex]\( (x^2 - 6x) \)[/tex] into [tex]\( (x + 3) \)[/tex]:
[tex]\[
= x^2(x + 3) - 6x(x + 3)
\][/tex]
Multiply each term:
[tex]\[
= x^3 + 3x^2 - 6x^2 - 18x = x^3 - 3x^2 - 18x
\][/tex]
So, [tex]\( (fg)(x) = (x + 3)(x^2 - 6x) \)[/tex] effectively matches the factored form.
(d) To find [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 6x}{x + 3}
\][/tex]
So, [tex]\( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 6x}{x + 3} \)[/tex].
Here is a summary of the results:
(a) [tex]\( (f+g)(x) = x^2 - 5x + 3 \)[/tex]
(b) [tex]\( (f-g)(x) = x^2 - 7x - 3 \)[/tex]
(c) [tex]\( (fg)(x) = (x + 3)(x^2 - 6x) \)[/tex]
(d) [tex]\( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 6x}{x + 3} \)[/tex]
