Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

If \(f(x)=\frac{x}{x-3}\), find \(f(-2), f(0)\), and \(f(3)\)

Step-by-Step Solution

Answer

\(f(-2) = \frac{2}{5}\), \(f(0) = 0\), \(f(3)\) is undefined.

Step 1: Substitute in the formula

Plug in the values of \(-2, 0,\) and \(3\) into the function \(f(x)=\frac{x}{x-3}\) to find \(f(-2)\), \(f(0)\), and \(f(3)\).

Step 2: Calculate f(-2)

Substitute \(-2\) into the function: \[f(-2) = \frac{-2}{-2-3} = \frac{-2}{-5} = \frac{2}{5}\].

Step 3: Calculate f(0)

Substitute \(0\) into the function: \[f(0) = \frac{0}{0-3} = \frac{0}{-3} = 0\].

Step 4: Evaluate f(3) (Check for undefined values)

Substitute \(3\) into the function to check if it is defined: \[f(3) = \frac{3}{3-3} = \frac{3}{0}\]. Since division by zero is undefined, \(f(3)\) is undefined.

Key Concepts

Function Evaluation

In mathematics, evaluating a function involves finding the output value of a function given an input value. For the function \(f(x) = \frac{x}{x-3}\), we want to find the output values for different inputs. Plug in the desired value for \(x\) into the function and simplify to find \(f(x)\). This process is called function evaluation. In practice:

• Replace \(x\) with the given input value.
• Simplify the resulting expression as much as possible.
• Determine the output value or identify if the expression is undefined.

For example, Function evaluation for \(f(x) = \frac{x}{x-3}\) with \(x = -2\) is:\[f(-2) = \frac{-2}{-2-3} = \frac{-2}{-5} = \frac{2}{5}\]This yields \(f(-2) = \frac{2}{5}\). Repeating similar steps for different inputs helps you understand how the function behaves.

Undefined Expressions

In the world of mathematics, some expressions appear that cannot be assigned a value. An undefined expression happens when an operation can't be completed within the logical rules of mathematics. One classic example is division by zero, which is not allowed. In our original exercise, if we attempt to evaluate the function \(f(x) = \frac{x}{x-3}\) at \(x = 3\), we find:

• Substitute \(x = 3\) into the expression.
• The denominator becomes \(0\), leading to division by zero: \[f(3) = \frac{3}{3-3} = \frac{3}{0}\]

Since division by zero is impossible, the expression is undefined, meaning \(f(3)\) does not exist in the real number system. Undefined expressions indicate limits of the function and can help identify function restrictions.

Division by Zero

A fundamental concept to understand with rational functions is the rule against dividing by zero. This rule is critical when evaluating functions that include a variable in the denominator, such as \(f(x) = \frac{x}{x-3}\). Division by zero is undefined because there is no number that can multiply with zero to produce a non-zero value. Hence, when a denominator equals zero, we can't perform the division.

• Identify parts of the function where the denominator equates to zero.
• These values are not part of the function's domain, i.e., where the function is undefined.
• In our example, \(x = 3\) results in \(x-3=0\) making \(f(x)\) undefined.

Understanding division by zero assists in analyzing where a rational function may have vertical asymptotes, indicating values excluded from the domain.