Problem 46 - Intermediate Algebra

Question

If \(f(x)=\frac{3}{x-2}\), find \(f(3), f(0), f(-1)\), and \(f(-5)\).

Step-by-Step Solution

Answer

\(f(3) = 3\), \(f(0) = -\frac{3}{2}\), \(f(-1) = -1\), \(f(-5) = -\frac{3}{7}\).

Step 1: Understanding the Function f(x)

The function given is \( f(x) = \frac{3}{x-2} \). This is a rational function where the numerator is 3 and the denominator is \( x-2 \). Our task is to substitute different values for \( x \) to find the function values \( f(3) \), \( f(0) \), \( f(-1) \), and \( f(-5) \).

Step 2: Finding f(3)

To find \( f(3) \), substitute \( x = 3 \) into the function: \( f(3) = \frac{3}{3-2} = \frac{3}{1} = 3 \). So, \( f(3) = 3 \).

Step 3: Finding f(0)

To find \( f(0) \), substitute \( x = 0 \) into the function: \( f(0) = \frac{3}{0-2} = \frac{3}{-2} = -\frac{3}{2} \). So, \( f(0) = -\frac{3}{2} \).

Step 4: Finding f(-1)

To find \( f(-1) \), substitute \( x = -1 \) into the function: \( f(-1) = \frac{3}{-1-2} = \frac{3}{-3} = -1 \). So, \( f(-1) = -1 \).

Step 5: Finding f(-5)

To find \( f(-5) \), substitute \( x = -5 \) into the function: \( f(-5) = \frac{3}{-5-2} = \frac{3}{-7} = -\frac{3}{7} \). So, \( f(-5) = -\frac{3}{7} \).

Key Concepts

Function Evaluation

Evaluating a function essentially means finding the output for a specific input value. In our case, given the function \(f(x) = \frac{3}{x-2}\), we need to find out what happens to the function when specific values replace \(x\). Function evaluation helps us understand how a function behaves with different inputs and is often one of the most straightforward tasks when dealing with functions.

• Start by identifying the given function.
• Then plug the given values into the function one at a time.
• Finally, perform the arithmetic to get the result for each evaluation.

This method ensures we can reliably determine the output, giving us valuable insights into the function's nature.

Substitution in Functions

Substitution is a key process in function evaluation. It involves replacing the variable \(x\) in the function with a specific number. This technique helps us explore how changes in input values affect the output.When substituting, follow these steps:

• Identify the point of substitution within the function. Here, the function is \(f(x) = \frac{3}{x-2}\).
• Replace \(x\) with a designated value like 3, 0, -1, or -5.
• Calculate the new expression to find the result.

It's like a math puzzle, where each piece replaced (or substituted) fits into a specific spot, eventually unveiling the complete picture.

Finding Function Values

Finding function values is the ultimate goal of substitution and evaluation. It's about calculating the numerical output of a function when given specific inputs. In the exercise, we sought to determine \(f(3)\), \(f(0)\), \(f(-1)\), and \(f(-5)\) by plugging these values into \(f(x) = \frac{3}{x-2}\).Here's a quick summary of how we found each value:

• For \(f(3)\), substitute 3: \(f(3) = \frac{3}{1} = 3\).
• For \(f(0)\), substitute 0: \(f(0) = \frac{3}{-2} = -\frac{3}{2}\).
• For \(f(-1)\), substitute -1: \(f(-1) = \frac{3}{-3} = -1\).
• For \(f(-5)\), substitute -5: \(f(-5) = \frac{3}{-7} = -\frac{3}{7}\).

Each result tells us precisely how the function responds to these inputs, allowing us to plot, analyze, and understand the broader behavior of the function.