Question

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

Short answer

Question: Determine the value of the function \(f(x) = \frac{1}{x^3 +1}\) at \(x=2\) and \(x=y^2\). Answer: When \(x=2\), the function's value is \(f(2) = \frac{1}{9}\), and when \(x=y^2\), the function's value is \(f(y^2) = \frac{1}{y^6 + 1}\).

Step-by-step solution

Evaluate the function at x = 2

To find the value of the function at \(x = 2\), we substitute \(x = 2\) into the function and calculate the result: \(f(2) = \frac{1}{2^3 + 1}\).

Calculate the value of the function at x = 2

Calculate the value of the expression \(\frac{1}{2^3 + 1}\): \(f(2) = \frac{1}{8 + 1} = \frac{1}{9}\). So, \(f(2) = \frac{1}{9}\).

Evaluate the function at x = y^2

To find the value of the function at \(x=y^2\), we substitute \(x = y^2\) into the function and simplify the result: \(f(y^2) = \frac{1}{(y^2)^3 + 1}\).

Simplify the function at x = y^2

Now, we simplify the expression \(\frac{1}{(y^2)^3 + 1}\): \(f(y^2) = \frac{1}{y^6 + 1}\). So, \(f(y^2) = \frac{1}{y^6 + 1}\). The answer is \(f(2) = \frac{1}{9}\) and \(f(y^2) = \frac{1}{y^6 + 1}\).

Key concepts

Evaluating Functions

Understanding how to evaluate functions is a foundational skill in algebra that allows you to find the value of a function for specific inputs. Evaluation involves substituting a given value into the function in place of the variable and performing the necessary calculations to simplify the expression. For instance, if you have a function defined as f(x) = 1/(x^3 + 1), evaluating this function at x = 2 means replacing every instance of x in the formula with 2.

The process follows these steps:

• Identify the input value.
• Replace the function's variable with the given input.
• Perform arithmetic operations following the order of operations.
• Simplify the resultant expression to find the output value.

Remember, parentheses are essential to maintain the correct order of operations while substituting the variable. Evaluating functions accurately is crucial for understanding how the function behaves with different inputs.

Simplifying Expressions

Simplifying expressions is about rewriting them in a more basic or compact form without changing their value. This involves tasks such as combining like terms, using exponent rules, and performing arithmetic operations. In the context of the exercise, simplifying the expression after substituting 2 into the function involved performing the operation 2^3 and then adding 1 to the result, which finally gave us f(2) = 1/9.

The steps in simplification often include:

• Applying exponent rules correctly (e.g., (y^2)^3 becomes y^6).
• Performing arithmetic operations such as addition, subtraction, multiplication, and division.
• Combining like terms if applicable.

Simplifying expressions makes them easier to understand and work with, especially when dealing with more complex functions or when graphing function behavior.

Substituting Variables

Substituting variables is a technique used to replace variables with numbers or other expressions in an algebraic context. When evaluating functions, it is often necessary to substitute one variable for another, or a number for a variable. This allows you to work with different forms of the function and to find specific values, as shown in the exercise.

To effectively substitute variables, follow these tips:

• Write down the function with the original variable.
• Clearly identify the substitution to be made and replace all instances of the variable.
• Ensure that the new expression is written correctly, respecting hierarchy and mathematical conventions.
• Simplify the resulting expression as needed to find the final form.

f(y^2) is the function f(x) after substituting x with y^2. The act of substituting not only assists in function evaluation but also in understanding how functions change with different forms of input.