Question

\(y=x^{2}-x\) and \(x=0\)

Short answer

The value of \(y\) is 0 when \(x=0\).

Step-by-step solution

Substitute 'x' in the Expression

To find the value of the function at a specific point, we substitute the given value of \(x\) into the equation. Here, we need to substitute \(x=0\) into the equation \(y=x^{2}-x\).

Simplify the Expression

Substitute \(x=0\) into the expression: \(y=0^2-0\). Simplifying this, we have \(y=0-0\), which results in \(y=0\).

Key concepts

Function Evaluation

Function evaluation is the process of finding the output of a function, given an input. In our example, the function equation is given as \(y=x^{2}-x\). This means that whatever value we choose for \(x\), the output \(y\) will be computed using this formula. For example, if we substitute \(x=0\), then:

• Replace every instance of \(x\) in the equation with \(0\).
• Calculate the value of the function by following the operations defined in the equation.

Function evaluation is a fundamental concept in calculus and is used to understand how functions behave at specific points. By evaluating functions at different values of \(x\), we can trace out the curve of the function on a graph.

Substitution Method

The substitution method in calculus involves replacing variables in a function with numbers to find specific output values, or with other expressions to simplify the problem. In our exercise, we have the expression \(y=x^{2}-x\) and we need to find the value of the function when \(x=0\).To use the substitution method:

• Take the equation \(y=x^{2}-x\).
• Replace \(x\) with \(0\), so that it becomes \(y=0^2-0\).
• This directly simplifies into a form that is easy to calculate.

The substitution method is powerful because it allows us to handle complex expressions by breaking them down into simpler, more manageable computations.

Simplifying Expressions

When simplifying expressions, the goal is to make an expression as simple as possible while maintaining its value. In our example, the result of substituting \(x=0\) into the function \(y=x^{2}-x\) gave us \(y=0^2-0\).Here's how to simplify:

• Calculate each term individually, \(0^2 = 0\) and \(-0 = 0\).
• Combine the terms to get \(y=0-0\).
• Simplify the result to \(y=0\).

By simplifying expressions, we reduce complex equations to their simplest form, making it easier to understand their outputs. This is essential not only for solving mathematical problems but also for interpreting the results accurately.