Let ƒ(x) = 1/ (x³+1).
Compute ƒ(2) and ƒ(y²).

Verified step by step guidance

1. Identify the function given: \( f(x) = \frac{1}{x^3 + 1} \).
2. To compute \( f(2) \), substitute \( x = 2 \) into the function: \( f(2) = \frac{1}{2^3 + 1} \).
3. Simplify the expression for \( f(2) \): calculate \( 2^3 \) and add 1.
4. To compute \( f(y^2) \), substitute \( x = y^2 \) into the function: \( f(y^2) = \frac{1}{(y^2)^3 + 1} \).
5. Simplify the expression for \( f(y^2) \): calculate \( (y^2)^3 \) and add 1.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, to compute ƒ(2) for the function ƒ(x) = 1/(x³ + 1), you replace x with 2, resulting in ƒ(2) = 1/(2³ + 1) = 1/9.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In the function ƒ(x) = 1/(x³ + 1), the denominator x³ + 1 is a polynomial of degree three, which influences the behavior and properties of the function.

Substitution in Functions

Substitution in functions refers to replacing a variable with another expression or value. In this case, computing ƒ(y²) means substituting y² into the function, leading to ƒ(y²) = 1/((y²)³ + 1) = 1/(y^6 + 1), which allows for further analysis of the function's behavior based on the variable y.