Question

State whether it’s a function or not

Answer 1

Answer:

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

Step-by-step explanation:

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

a) Use the Horizontal Line Test to decide whether $f$ has an inverse function.

As shown in the graph below there is no any horizontal line that intersects the graph of $f$ 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 $f(x)$ by $y$ in the equation for $f(x)$.

$y=\frac{1}{x^3}$

c) Interchange the roles of $x$ and $y$ and solve for $y$

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

d) Replace $y$ by $f^{-1}(x)$ in the new equation.

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

So the correct option is:

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

Answer 2

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.