Answer:
[tex]\left(\dfrac{f}{g}\right)(x) = -\dfrac{1}{2x^{4/3}}[/tex]
Step-by-step explanation:
12)
To find [tex]\left(\dfrac{f}{g}\right)(x)[/tex], we need to divide [tex]f(x)[/tex] by [tex]g(x)[/tex].
Given that:
- [tex]f(x) = -2x^{2/3}[/tex]
- [tex]g(x) = 4x^2[/tex]
We can express [tex]\left(\dfrac{f}{g}\right)(x)[/tex] as:
[tex]\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}[/tex]
Now, let's substitute the given functions into the expression:
[tex]\left(\dfrac{f}{g}\right)(x) = \dfrac{-2x^{2/3}}{4x^2}[/tex]
To simplify, we can divide both the numerator and the denominator by [tex]2x^{2/3}[/tex] since it's a common factor:
[tex]\left(\dfrac{f}{g}\right)(x) = \dfrac{-\dfrac{2}{2}x^{2/3}}{\dfrac{4}{2}x^2}[/tex]
This simplifies to:
[tex]\left(\dfrac{f}{g}\right)(x) = \dfrac{-x^{2/3}}{2x^2}[/tex]
Now, when we divide [tex]x^{2/3}[/tex] by [tex]x^2[/tex], we subtract the exponents:
[tex]\left(\dfrac{f}{g}\right)(x) = -\dfrac{1}{2x^{2 - 2/3}}[/tex]
[tex]\left(\dfrac{f}{g}\right)(x) = -\dfrac{1}{2x^{6/3 - 2/3}}[/tex]
[tex]\left(\dfrac{f}{g}\right)(x) = -\dfrac{1}{2x^{4/3}}[/tex]
Therefore, [tex]\left(\dfrac{f}{g}\right)(x) = -\dfrac{1}{2x^{4/3}}[/tex].
