Problem 72 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=\frac{x-2}{x+3}\)

Step-by-Step Solution

Answer

\(f(-2) = -4\), \(f(-1) = -1.5\), \(f(0) = -\frac{2}{3}\), \(f(1) = -\frac{1}{4}\), \(f(2) = 0\).

Step 1: Evaluate f(-2)

To find \(f(-2)\), substitute \(-2\) for \(x\) in the function: \[f(-2) = \frac{-2-2}{-2+3} = \frac{-4}{1} = -4.\] So, \(f(-2) = -4\).

Step 2: Evaluate f(-1)

To find \(f(-1)\), substitute \(-1\) for \(x\) in the function: \[f(-1) = \frac{-1-2}{-1+3} = \frac{-3}{2} = -1.5.\] So, \(f(-1) = -1.5\).

Step 3: Evaluate f(0)

To find \(f(0)\), substitute \(0\) for \(x\) in the function: \[f(0) = \frac{0-2}{0+3} = \frac{-2}{3} = -\frac{2}{3}.\] So, \(f(0) = -\frac{2}{3}\).

Step 4: Evaluate f(1)

To find \(f(1)\), substitute \(1\) for \(x\) in the function: \[f(1) = \frac{1-2}{1+3} = \frac{-1}{4} = -\frac{1}{4}.\] So, \(f(1) = -\frac{1}{4}.\)

Step 5: Evaluate f(2)

To find \(f(2)\), substitute \(2\) for \(x\) in the function: \[f(2) = \frac{2-2}{2+3} = \frac{0}{5} = 0.\] So, \(f(2) = 0.\)

Key Concepts

Function Substitution

Function substitution is the process of replacing the variable in a function with a specific value to find the function's output at that particular instance. Essentially, you're solving the function for a given input.

When working with a function like \( f(x) = \frac{x-2}{x+3} \), evaluating it for different values of \( x \) means plugging these values into the function and simplifying.

• Identify the variable in your function, which is \( x \) in our case.
• Substitute the given value for \( x \) into the function.
• Simplify the expression to get the result.

For example, to evaluate \( f(-2) \): substitute \( -2 \) for \( x \), perform the arithmetic operations, and simplify, leading to \( f(-2) = -4 \). With practice, this straightforward approach makes solving functions much easier.

Rational Functions

Rational functions are a type of function represented by the ratio of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are polynomials.

The function \( f(x) = \frac{x-2}{x+3} \) is a rational function. Note:

• The numerator is \( x-2 \).
• The denominator is \( x+3 \).

Rational functions have some unique characteristics:

• They can have vertical asymptotes, where the function goes to infinity, occurring where the denominator is zero and the function is undefined.
• They might have horizontal or slant asymptotes, affecting end behavior.

In our example, \( f(x) \) is undefined at \( x = -3 \) because dividing by zero isn't possible. So, always check the denominator in rational functions.

Algebraic Operations

Understanding algebraic operations, like addition, subtraction, multiplication, and division, is crucial for handling functions efficiently. These operations help simplify expressions and evaluate functions accurately.When evaluating functions like \( f(x) = \frac{x-2}{x+3} \):

• Start by substituting the value into the function.
• Perform the operations in the numerator and the denominator separately.
• Simplify the fraction, which often involves arithmetic operations.

For instance, using \( f(1) \):- Substitute \( 1 \) for \( x \) in the numerator, \( 1-2 = -1 \).- Do the same in the denominator, \( 1+3 = 4 \).- Simplify \( \frac{-1}{4} \).By mastering basic arithmetic and algebraic operations, simplifying and evaluating any function becomes a manageable task.