Question

Find each of the following functions and state their domains. (Enter the domains in interval notation.)
f(x) = x3 + 4x2, g(x) = 5x2 − 1
(a) f + g
f + g =
domain =
(b) f - g
f − g =
domain =
(c) fg
fg =
domain =
(d) f/g
f/g =
domain =

Answer 1

Answer:

Step-by-step explanation:

Given functions are,

f(x) = x³ + 4x²

g(x) = 5x² - 1

(a). (f + g)(x) = f(x) + g(x)

                   = x³ + 4x² + 5x² - 1

                   = x³ + 9x² - 1

   Since the composite function is defined for all values of x, domain of the function will be (-∞, ∞)

(b). (f - g)(x) = x³ + 4x² - (5x² - 1)

                 = x³ - x² + 1

  Domain of the function : (-∞, ∞)

(c). (f.g)(x) = f(x) × g(x)

               = (x³ + 4x²)(5x² - 1)

               = 5x⁵+ 20x⁴ - x³ - 4x²

Domain of the function will be (-∞, ∞).

(d). $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$

               = $\frac{x^3+4x^2}{5x^2-1}$

Since this function is not defined at [5x² - 1 = 0] Or x = ±$\frac{1}{\sqrt{5}}$

Therefore, Domain of the composite function will be,

(-∞, -$\frac{1}{\sqrt{5}}$) ∪ $(-\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}})$ ∪ ($\frac{1}{\sqrt{5}}$, ∞)