Respuesta :

danielmaduroh

Answer:

[tex]\boxed{c. \ f^{-1}(x)=\frac{1}{\sqrt[3]{x}}}[/tex]

Step-by-step explanation:

Let's find the inverse function of [tex]f(x)=\frac{1}{x^3}[/tex] to know what item we must choose as correct. So let's apply this steps:

a) Use the Horizontal Line Test to decide whether [tex]f[/tex] has an inverse function.

As shown in the graph below there is no any horizontal line that intersects the graph of [tex]f[/tex] at more than one point. Thus, the function is one-to-one and has an inverse function. Therefore, the inverse is a function..

b) Replace [tex]f(x)[/tex] by [tex]y[/tex] in the equation for [tex]f(x)[/tex].

[tex]y=\frac{1}{x^3}[/tex]

c) Interchange the roles of [tex]x[/tex] and [tex]y[/tex] and solve for [tex]y[/tex]

[tex]x=\frac{1}{y^3} \\ \\ \therefore y^3=\frac{1}{x} \\ \\ \therefore y=\frac{1}{\sqrt[3]{x}}[/tex]

d) Replace [tex]y[/tex] by [tex]f^{-1}(x)[/tex] in the new equation.

[tex]f^{-1}(x)=\frac{1}{\sqrt[3]{x}}[/tex]

So the correct option is:

[tex]c. \ f^{-1}(x)=\frac{1}{\sqrt[3]{x}}[/tex]

zainebamir540

Answer:

Choice C is correct.

Step-by-step explanation:

We have given a function :

f(x) = 1/x³

We have to find the inverse of function.

let y = f(x) we get,

y = 1/x³

Replacing x and y we get,

x = 1/y³

y³ = 1/x

Taking third roots on both sides we get,

y = 1/∛x

Replacing y to f⁻¹ (x)  we get,

f⁻¹ (x)= 1/∛x is the inverse of f(x) = 1/x³.

So, As it  is one to one and inverse of function exist so it is a function.

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