State whether f is a function

A function [tex]f[/tex] from a set [tex]A[/tex] to a set [tex]B[/tex] is a relation that assigns to each element [tex]x[/tex] in the set [tex]A[/tex] exactly one element [tex]y[/tex] in the set [tex]B[/tex]. The set [tex]A[/tex] is the domain (also called the set of inputs) of the function and the set [tex]B[/tex] 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 [tex]f[/tex] has an inverse function if and only if there is no any horizontal line that intersects the graph of [tex]f[/tex] at more than one point. So the function is called one-to-one. The graph of [tex]f[/tex] 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 [tex]f^-{1}(x)[/tex]:

[tex]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}}}[/tex]

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.

zainebamir540 zainebamir540 · Answer 2 · 2019-03-10T13:57:27+01:00

Answer:

[tex](\frac{y}{6 })^{1/4}=f^{-1}(y)[/tex]

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

[tex](\frac{y}{6})^{1/4}=x[/tex]

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

[tex](\frac{y}{6 })^{1/4}=f^{-1}(y)[/tex]

Replacing y by x , we have

[tex](\frac{±x}{6 })^{1/4}=f^{-1}(x)[/tex] 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.