Question

Find, then state if it’s a function

Answer 1

Answer:

The answer is y = ± $\sqrt[4]{\frac{x}{4}}$ ,  $f^{-1}(x)$ is a function ⇒ 1st answer

Step-by-step explanation:

Let us revise the steps of find the inverse of a function

1. Replace f(x) with y
2. Replace every x by y and replace y by x
3. Solve the equation in Step 2 for y
4. Replace y by $f^{-1}(x)$

∵ $f(x)=4x^{4}$

- Replace f(x) by y

∴ $y=4x^{4}$

- Replace y by x and x by y

∴ $x=4y^{4}$

- Divide each side by 4

∴ $\frac{x}{4}=y^{4}$

- Take $\sqrt[4]{}$  for both sides

∴ ± $\sqrt[4]{\frac{x}{4}}=y$

- Switch the two sides

∴ y = ± $\sqrt[4]{\frac{x}{4}}$

∵ There is no fourth root for negative number

∴ x ≥ 0

When you test the graph of $f^{-1}(x)$ 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

∴  $f^{-1}(x)$ is a function

The answer is y = ± $\sqrt[4]{\frac{x}{4}}$ ,  $f^{-1}(x)$ is a function