Problem 47 - Intermediate Algebra

Question

If \(f(x)=\frac{-4}{x+3}\), find \(f(1), f(-1), f(3)\), and \(f(-6)\).

Step-by-Step Solution

Answer

\(f(1) = -1\), \(f(-1) = -2\), \(f(3) = -\frac{2}{3}\), \(f(-6) = \frac{4}{3}\).

Step 1: Understanding the Function

The function given is \(f(x) = \frac{-4}{x+3}\). This function represents a rational equation where \(x\) cannot be \(-3\) because it would make the denominator zero.

Step 2: Calculate \(f(1)\)

Substitute \(x = 1\) into the function: \[f(1) = \frac{-4}{1+3} = \frac{-4}{4} = -1\] So, \(f(1) = -1\).

Step 3: Calculate \(f(-1)\)

Substitute \(x = -1\) into the function: \[f(-1) = \frac{-4}{-1+3} = \frac{-4}{2} = -2\] So, \(f(-1) = -2\).

Step 4: Calculate \(f(3)\)

Substitute \(x = 3\) into the function: \[f(3) = \frac{-4}{3+3} = \frac{-4}{6} = -\frac{2}{3}\] So, \(f(3) = -\frac{2}{3}\).

Step 5: Calculate \(f(-6)\)

Substitute \(x = -6\) into the function: \[f(-6) = \frac{-4}{-6+3} = \frac{-4}{-3} = \frac{4}{3}\] So, \(f(-6) = \frac{4}{3}\).

Key Concepts

Function Evaluation

Function evaluation is the process where we plug specific values into a function and solve to find the corresponding outputs. This is applicable for any type of function, including rational functions. When evaluating a function, you replace the variable (usually represented by "x") in the function's formula with the number you are given.In the exercise, we have the function, \(f(x) = \frac{-4}{x+3}\), and we're asked to find \(f(1)\), \(f(-1)\), \(f(3)\), and \(f(-6)\). Here's how it works:- Step 1: Identify the function you're working with.- Step 2: Replace the variable in the formula with the given number.- Step 3: Simplify the expression to find the result.Through this method, you can determine the outputs, or function values, for particular inputs by substituting these inputs into the given function.

Undefined Values

Understanding undefined values in rational functions is crucial as it prevents making mathematical errors. A rational function like \(f(x) = \frac{-4}{x+3}\) involves a fraction with a denominator. For any fraction, a denominator of zero makes the expression undefined because division by zero is not possible.Key points to remember:- Identify the denominator in your rational function.- Set the denominator equal to zero and solve for "x" to find points where the function is undefined.- For \(f(x) = \frac{-4}{x+3}\), the denominator \(x + 3 = 0\) leads to \(x = -3\). So, at \(x = -3\), the function is undefined.Avoiding undefined values ensures that the function behaves as expected and allows you to evaluate it properly for other values of \(x\). By understanding where a function is undefined, you can make informed choices about which inputs to use during evaluation.

Algebraic Substitution

Algebraic substitution is a method used to simplify expressions by replacing a variable with a given number. This process is particularly helpful in evaluating functions such as the rational function \(f(x) = \frac{-4}{x+3}\).Here's how algebraic substitution works:1. **Insert the Value**: Replace "x" with the given number. - For example, to find \(f(1)\), replace "x" with 1 in \(\frac{-4}{x+3}\).2. **Simplify the Expression**: Carry out basic arithmetic to simplify the equation. After substitution, the expression for \(f(1)\) becomes \(\frac{-4}{1+3} = \frac{-4}{4} = -1\).3. **Repeat as Needed**: For each different value, repeat the substitution process.This systematic approach makes it easy to derive the answer without confusion. Algebraic substitution simplifies the function to a numerical result, offering clear pathways to solutions through consistent application.