Question

What is (f-g)(x)?
F(x)=3x5+6x2-5
G(x)=3x5-5x4+3x2-15

What is (f•g)(x)?
F(x)=x3-4x+2
G(x)=x2-6

Answer 1

The resulting function is (f-g)(x) = 5 · x⁴ + 3 · x² + 10.

The resulting function is (f • g) (x) = 3 · (3 · x⁵ - 5 · x⁴ + 3 · x² - 15)⁵ + 6 · (3 · x⁵ - 5 · x⁴ + 3 · x² -15)² - 5.  

How to apply subtraction between two functions

Let be $f(x)$ and $g(x)$ polynomial functions, the subtraction between between the two functions implies the subtraction of similar terms. If we know that $f(x) = 3\cdot x^{5}+6\cdot x^{2}-5$ and $g(x) = 3\cdot x^{5}-5\cdot x^{4}+3\cdot x^{2}-15$, then the subtraction between the two functions:

$(f-g)(x) = (3\cdot x^{5}+6\cdot x^{2}-5)-(3\cdot x^{5}-5\cdot x^{4}+3\cdot x^{2}-15)$

$(f-g)(x) = 5\cdot x^{4}+3\cdot x^{2}+10$

The resulting function is (f-g)(x) = 5 · x⁴ + 3 · x² + 10. $\blacksquare$

How to apply composition between two functions

The composition between two functions, the independent variable of the former function by the latter function. If we know that $f(x) = 3\cdot x^{5}+6\cdot x^{2}-5$ and $g(x) = 3\cdot x^{5}-5\cdot x^{4}+3\cdot x^{2}-15$, then the composition between the two functions:

$f\,\circ\,g\,(x) = 3\cdot (3\cdot x^{5}-5\cdot x^{4}+3\cdot x^{2}-15)^{5}+6\cdot (3\cdot x^{5}-5\cdot x^{4}+3\cdot x^{2}-15)^{2}-5$

(f • g) (x) = 3 · (3 · x⁵ - 5 · x⁴ + 3 · x² - 15)⁵ + 6 · (3 · x⁵ - 5 · x⁴ + 3 · x² -15)² - 5.  $\blacksquare$

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