# Solved Examples on the Arithmetic Operations on Functions

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(1.)

$f(x) = x + 2 \\[3ex] g(x) = x^2 - x \\[3ex]$ Calculate $(f + g)(-5)$ using at least two methods.

First Method (long way)
$(f + g)(x) \\[3ex] = f(x) + g(x) \\[3ex] = (x + 2) + (x^2 - x) \\[3ex] = x + 2 + x^2 - x \\[3ex] = x^2 + 2 \\[3ex] (f + g)(x) = x^2 + 2 \\[3ex] (f + g)(-5) \\[3ex] = ({-5})^2 + 2 \\[3ex] = 25 + 2 \\[3ex] = 27 \\[3ex]$ Second Method (short way)
$(f + g)(-5) \\[3ex] = f(-5) + g(-5) \\[3ex] f(x) = x + 2 \\[3ex] f(-5) = -5 + 2 \\[3ex] = -3 \\[3ex] g(x) = x^2 - x \\[3ex] g(-5) = ({-5})^2 - (-5) \\[3ex] = 25 + 5 \\[3ex] = 30 \\[3ex] f(-5) + g(-5) \\[3ex] = -3 + 30 \\[3ex] = 27$
(2.)

$f(x) = 10x^2 - 2x \\[3ex] g(x) = 2x \\[3ex]$ Calculate:
(a.) $(f + g)(x)$ and the domain of the sum
(b.) $(f - g)(x)$ and the domain of the difference
(c.) $(fg)(x)$ and the domain of the product
(d.) $\left(\dfrac{f}{g}\right)(x)$ and the domain of the quotient

$(f + g)(x) = f(x) + g(x) \\[3ex] = (10x^2 - 2x) + 2x \\[3ex] = 10x^2 - 2x + 2x \\[3ex] = 10x^2 \\[3ex] D = (-\infty, \infty) \\[7ex] (f - g)(x) = f(x) - g(x) \\[3ex] = (10x^2 - 2x) - 2x \\[3ex] = 10x^2 - 2x - 2x \\[3ex] = 10x^2 - 4x \\[3ex] D = (-\infty, \infty) \\[7ex] (f * g)(x) = f(x) * g(x) \\[3ex] = (10x^2 - 2x) * 2x \\[3ex] = 2x(10x^2 - 2x) \\[3ex] = 20x^3 - 4x^2 \\[3ex] D = (-\infty, \infty) \\[7ex] \left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)} \\[5ex] = (10x^2 - 2x) \div 2x \\[3ex] = \dfrac{10x^2 - 2x}{2x} \\[5ex] Factor\:\: by\:\: GCF \\[3ex] GCF = 2x \\[2ex] = \dfrac{2x(5x - 1)}{2x} \\[5ex] = 5x - 1 \\[3ex] D = (-\infty, \infty)$
(3.)

$f(x) = x^2 - 1 \\[3ex] g(x) = 18 - x \\[3ex]$ Calculate $(f - g)(-4)$ using at least two methods.

First Method (long way)
$(f - g)(x) \\[3ex] = f(x) - g(x) \\[3ex] = (x^2 - 1) - (18 - x) \\[3ex] = x^2 - 1 - 18 + x \\[3ex] = x^2 + x - 19 \\[3ex] (f - g)(x) = x^2 - x + 17 \\[3ex] (f - g)(-4) \\[3ex] = (-4)^2 + (-4) - 19 \\[3ex] = 16 - 4 - 19 \\[3ex] = -7 \\[3ex]$ Second Method (short way)
$(f - g)(-4) \\[3ex] = f(-4) - g(-4) \\[3ex] f(x) = x^2 - 1 \\[3ex] f(-4) = {-4}^2 - 1 \\[3ex] = 16 - 1 \\[3ex] = 15 \\[3ex] g(x) = 18 - x \\[3ex] g(-4) = 18 - (-4) \\[3ex] = 18 + 4 \\[3ex] = 22 \\[3ex] f(-4) - g(-4) \\[3ex] = 15 - 22 \\[3ex] = -7$
(4.)

$f(x) = \dfrac{4}{x - 3} \\[5ex] g(x) = \sqrt{x - 2} \\[3ex]$ Calculate:
(a.) $(f + g)(x)$ and the domain of the sum
(b.) $(f - g)(x)$ and the domain of the difference
(c.) $(fg)(x)$ and the domain of the product
(d.) $\left(\dfrac{f}{g}\right)(x)$ and the domain of the quotient

$(f + g)(x) = f(x) + g(x) \\[3ex] = \dfrac{4}{x - 3} + \sqrt{x - 2} \\[5ex] For\:\: \dfrac{4}{x - 3}; D = (-\infty, 3) \cup (3, \infty) \\[5ex] For\:\: \sqrt{x - 2}; D = [2, \infty) \\[3ex]$ Find the domain that is common to the sum of both functions
Domain of the sum, $D = [2, 3) \cup (3, \infty)$

$(f - g)(x) = f(x) - g(x) \\[3ex] = \dfrac{4}{x - 3} - \sqrt{x - 2} \\[5ex] For\:\: \dfrac{4}{x - 3}; D = (-\infty, 3) \cup (3, \infty) \\[5ex] For\:\: \sqrt{x - 2}; D = [2, \infty) \\[3ex]$ Find the domain that is common to the difference of both functions
Domain of the difference, $D = [2, 3) \cup (3, \infty)$

$(f * g)(x) = f(x) * g(x) \\[3ex] = \dfrac{4}{x - 3} * \sqrt{x - 2} \\[3ex] = \dfrac{4\sqrt{x - 2}}{x - 3} \\[3ex] For\:\: the\:\: denominator; D = (-\infty, 3) \cup (3, \infty) \\[3ex] For\:\: the\:\: numerator; D = [2, \infty) \\[3ex]$ Find the intersection of both domains
Domain of the product, $D = [2, 3) \cup (3, \infty)$

$\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)} \\[5ex] = \dfrac{4}{x - 3} \div \sqrt{x - 2} \\[5ex] = \dfrac{4}{x - 3} * \dfrac{1}{\sqrt{x - 2}} \\[5ex] = \dfrac{4}{(x - 3)\sqrt{x - 2}} \\[5ex] = \dfrac{4}{(x - 3)\sqrt{x - 2}} * \dfrac{\sqrt{x - 2}}{\sqrt{x - 2}} \\[5ex] = \dfrac{4\sqrt{x - 2}}{(x - 3)(x - 2)} \\[5ex] For\:\: the\:\: denominator; D = (-\infty, 2) \cup (2, 3) \cup (3, \infty) \\[3ex] For\:\: the\:\: numerator; D = [2, \infty) \\[3ex]$ Find the intersection of both domains
Domain of the quotient, $D = (2, 3) \cup (3, \infty)$
(5.) Perform the indicated operations on these functions.

$(a.)\;\; (g - f)(a) \;\;if \\[3ex] g(a) = -3a - 3 \\[3ex] f(a) = a^2 + 5 \\[5ex] (b.)\;\; (g - f)(x) \;\;if \\[3ex] g(x) = -x^2 - 1 - 2x \\[3ex] f(x) = x + 5 \\[5ex] (c.)\;\; (f \cdot g)(x) \;\;if \\[3ex] f(x) = 2x^3 - 5x^2 \\[3ex] g(x) = 2x - 1 \\[3ex]$

$(a.) \\[3ex] g(a) = -3a - 3 \\[3ex] f(a) = a^2 + 5 \\[3ex] (g - f)(a) \\[3ex] = g(a) - f(a) \\[3ex] = (-3a - 3) - (a^2 + 5) \\[3ex] = -3a - 3 - a^2 - 5 \\[3ex] = -a^2 - 3a - 8 \\[5ex] (b.) \\[3ex] g(x) = -x^2 - 1 - 2x \\[3ex] f(x) = x + 5 \\[3ex] (g - f)(x) \\[3ex] = g(x) - f(x) \\[3ex] = (-x^2 - 1 - 2x) - (x + 5) \\[3ex] = -x^2 - 2x - 1 - x - 5 \\[3ex] = -x^2 - 3x - 6 \\[5ex] (c.) \\[3ex] f(x) = 2x^3 - 5x^2 \\[3ex] g(x) = 2x - 1 \\[3ex] (f \cdot g)(x) \\[3ex] = f(x) \cdot g(x) \\[3ex] = (2x^3 - 5x^2)(2x - 1) \\[3ex] = 4x^4 - 2x^3 - 10x^3 + 5x^2 \\[3ex] = 4x^4 - 12x^3 + 5x^2$
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