Answer:
[tex]f^{-1}(x) = \sqrt[3]{x+5}[/tex]
Step-by-step explanation:
A function [tex]f^{-1}[/tex] is the inverse of f if whenever y =f(x) and [tex]x =f^{-1}[/tex]
To find the inverse of f(x) = [tex]x^{3}[/tex] - 5, we use the following steps:
- Step 1 : Put y for f(x) and solve for x
y = [tex]x^{3}[/tex] - 5
<=> [tex]x^{3}[/tex] = y + 5
<=> x = [tex]\sqrt[3]{y+ 5}[/tex]
- Step 2: Put [tex]f^{-1}(y)[/tex] for x, we have:
[tex]f^{-1}(y) =\sqrt[3]{y+5}[/tex]
- Step 3: Interchange y =x, we have
[tex]f^{-1}(x) = \sqrt[3]{x+5}[/tex]
Let Calculate 3 points on f(x)
x = 0 => y = -5
x = 1 => y = -4
x = 2 => y = 3
Let Calculate 3 points on [tex]f^{-1}[/tex] (x)
x = -5 => y = 0
x = -4 => y = 1
x = 3 => y = 2
Yes, the inverse is correct because:
- the domain of the inverse function is the range of the original function
- the range of the inverse function is the domain of the original function
