Answer:
The answer is y = ± [tex]\sqrt[4]{\frac{x}{4}}[/tex] , [tex]f^{-1}(x)[/tex] is a function ⇒ 1st answer
Step-by-step explanation:
Let us revise the steps of find the inverse of a function
- Replace f(x) with y
- Replace every x by y and replace y by x
- Solve the equation in Step 2 for y
- Replace y by [tex]f^{-1}(x)[/tex]
∵ [tex]f(x)=4x^{4}[/tex]
- Replace f(x) by y
∴ [tex]y=4x^{4}[/tex]
- Replace y by x and x by y
∴ [tex]x=4y^{4}[/tex]
- Divide each side by 4
∴ [tex]\frac{x}{4}=y^{4}[/tex]
- Take [tex]\sqrt[4]{}[/tex] for both sides
∴ ± [tex]\sqrt[4]{\frac{x}{4}}=y[/tex]
- Switch the two sides
∴ y = ± [tex]\sqrt[4]{\frac{x}{4}}[/tex]
∵ There is no fourth root for negative number
∴ x ≥ 0
When you test the graph of [tex]f^{-1}(x)[/tex] by a vertical line, it will cut it just at one in every position, so it is a function. Look to the attached graph for more understand
∴ [tex]f^{-1}(x)[/tex] is a function
The answer is y = ± [tex]\sqrt[4]{\frac{x}{4}}[/tex] , [tex]f^{-1}(x)[/tex] is a function
