Respuesta :
Answer:
[tex]f(x)=\sqrt[3]{x} +5\\\\\implies y = \sqrt{3}{x}+5[/tex]
a) Since the function is given to be one - one so the inverse of the function exist. Now f(x) maps x to y so the inverse of f(x) maps y to x
To find inverse, first interchange the roles of x and y :
[tex]\implies x = \sqrt[3]{y}+5[/tex]
Now, solve for y :
[tex]x = \sqrt[3]{y}+5\\\\\implies \sqrt[3]{y}=x-5\\\\\text{Now, cubing both the sides. We get,}\\\\\implies y=(x-5)^3\\\\\implies y=x^3-15\cdot x^2+75\cdot x-125\\\\\implies\bf f^{-1}(x)=x^3-15\cdot x^2+75\cdot x-125[/tex]
b) To find coordinates of f(x) :
[tex]f(x)=\sqrt[3]{x} +5[/tex]
First take y = 0 then take x = 0
⇒ x-coordinates : (-125,0) and y-coordinates : (0,5)
To find coordinates of inverse function of x :
[tex]y=x^3-15\cdot x^2+75\cdot x-125[/tex]
First take y = 0 then take x = 0
⇒ x - coordinates : (5,0) and y - coordinates : (0,-125)
c) f(x) is defined for every real number.
⇒ Domain : -∞ < x < ∞
and Range : -∞ < f(x) < ∞
And inverse function of x is also defined for every real number :
⇒ Domain : -∞ < x < ∞
[tex]\text{and Range : }-\infty < f^{-1}(x) < \infty [/tex]
