Question

State whether f is a function

Answer 1

Answer:

$y=\pm (\frac{x}{6})^{\frac{1}{4}} \ is \ not \ a \ function$

Step by step solution:

A function $f$ from a set $A$ to a set $B$ is a relation that assigns to each element $x$ in the set $A$ exactly one element $y$ in the set $B$. The set $A$ is the domain (also called the set of inputs) of the function and the set $B$ contains the range (also called the set of outputs). On the other hand, a function has an inverse function if and only if passes the Horizontal Line Test for Inverse Functions. This test tells us that a function $f$ has an inverse function if and only if there is no any horizontal line that intersects the graph of $f$ at more than one point. So the function is called one-to-one. The graph of $f$ is shown below. As you can see, this function does not pass the Horizontal Line Test, therefore the inverse is not a function. However, let's find $f^-{1}(x)$:

$f(x)=6x^4 \\ \\ Substitute \ f(x) \ by \ y \\ \\ y=6x^4 \\ \\ Interchange \ x \ and \ y: \\ \\ x=6y^4 \\ \\ Solve \ for \ y: \\ \\ y^4=\frac{x}{6} \\ \\ Solving \\ \\y=\pm \sqrt[4]{\frac{x}{6}} \\ \\ \boxed{y=\pm \left(\frac{x}{6}\right)^{\frac{1}{4}}}$

and this is not a function because there are elements in the set of inputs that match with two elements in the set of outputs.

Answer 2

Answer:

$(\frac{y}{6 })^{1/4}=f^{-1}(y)$

Step-by-step explanation:

We have given a function.

f(x) = 6x⁴

We have to find the inverse of the given function.

Putting y=f(x) in given equation ,we have

y = 6x⁴

y/6 = x⁴

Taking 4th root to both sides of Above equation, we have

$(\frac{y}{6})^{1/4}=x$

Putting x = f⁻¹(y) in above equation, we have

$(\frac{y}{6 })^{1/4}=f^{-1}(y)$

Replacing y by x , we have

$(\frac{±x}{6 })^{1/4}=f^{-1}(x)$ which is the answer.

f⁻¹(x) is not function because it assigns each value of x to two values of y.

Choice D is correct answer.