Question

If \(f(x)=1 /\left(x^{3}+1\right),\) what is \(f(2) ?\) What is \(f\left(y^{2}\right) ?\)

Short answer

Answer: f(2) = 1/9, and f(y²) = 1/(y^6+1).

Step-by-step solution

Finding f(2)

First, we'll find the value of the function f(x) when x=2. We have: $$f(x) = \frac{1}{x^3+1}$$ Now, substitute x=2 into the function: $$f(2) = \frac{1}{(2)^3+1}$$

Simplifying f(2)

Now, we'll simplify the expression we obtained in Step 1: $$f(2) = \frac{1}{8+1}$$ $$f(2) = \frac{1}{9}$$ Thus, the value of f(2) is \(\frac{1}{9}\).

Finding f(y²)

Next, we'll find the value of the function f(x) in terms of y². Substitute x with y² into the function: $$f(y^2) = \frac{1}{(y^2)^3+1}$$

Simplifying f(y²)

Now, we'll simplify the expression we obtained in Step 3: $$f(y^2) = \frac{1}{y^6+1}$$ Thus, the value of f(y²) is \(\frac{1}{y^6+1}\).