Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\frac{1}{x-6}, \quad g(x)=\frac{7}{x}+6\)

Step-by-Step Solution

Answer

\( f(g(x)) = \frac{x}{7} \) and \( g(f(x)) = 7x - 36 \).

Step 1: Find \( f(g(x)) \)

To find \( f(g(x)) \), substitute \( g(x) = \frac{7}{x} + 6 \) into \( f(x) = \frac{1}{x-6} \). This gives us \( f(g(x)) = \frac{1}{\left(\frac{7}{x}+6\right) - 6} \).

Step 2: Simplify \( f(g(x)) \)

In the expression \( \frac{1}{\left(\frac{7}{x} + 6\right) - 6} \), the terms \(6\) and \(-6\) cancel each other, leaving us with \( \frac{1}{\frac{7}{x}} \). Simplify this by multiplying numerator and denominator by \(x\), resulting in \( \frac{x}{7} \).

Step 3: Find \( g(f(x)) \)

To find \( g(f(x)) \), substitute \( f(x) = \frac{1}{x-6} \) into \( g(x) = \frac{7}{x} + 6 \). This results in \( g(f(x)) = \frac{7}{\left(\frac{1}{x-6}\right)} + 6 \).

Step 4: Simplify \( g(f(x)) \)

Simplify the expression \( \frac{7}{\left(\frac{1}{x-6}\right)} + 6 \) by recognizing that \( \frac{7}{\left(\frac{1}{x-6}\right)} = 7(x-6) \). Expanding this gives \( 7x - 42 + 6 \). Combine like terms to yield \( 7x - 36 \).

Key Concepts

Understanding Function Operations

When you perform operations on functions like adding, subtracting, multiplying, or dividing, you're engaging in what's known as "function operations." However, the composite functions exercise involves plugging one function into another, specifically evaluating \(f(g(x))\) and \(g(f(x))\). This process is essentially function substitution.

To find \(f(g(x))\), you replace each occurrence of the variable \(x\) in \(f(x)\) with the expression for \(g(x)\). Similarly, for \(g(f(x))\), you replace \(x\) in \(g(x)\) with \(f(x)\).

Composite functions can drastically change the behavior of the original functions, and understanding how to combine them is crucial. Always pay attention to the order: \(f(g(x))\) can differ from \(g(f(x))\). Keeping track of said transformations helps in forecasting the resulting function's behavior.

Exploring Rational Functions

A rational function is essentially a ratio of two polynomials, expressed as \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomial expressions. The functions in the exercise are examples of rational functions.

The function \(f(x)=\frac{1}{x-6}\) signifies that the variable \(x\) can’t be 6 because it would make the denominator zero, leading to an undefined function. On the other hand, \(g(x)=\frac{7}{x}+6\) shows restrictions wherever \(x=0\).

• Always identify the domain restrictions early - they are key to preventing errors.
• Polynomials in the numerator and denominator determine how the function behaves, and where it may be undefined.

Understanding the behavior and restrictions of these rational functions is vital for simplifying expressions like \(f(g(x))\) or \(g(f(x))\) correctly.

Mastering Simplifying Expressions

Simplifying complex expressions is like solving a puzzle: you're trying to make a complex expression as understandable as possible without changing its value.

Take, for example, the step of simplifying the composite function \(f(g(x)) = \frac{1}{\frac{7}{x}}\). Here, the key is recognizing that the fraction \(\frac{1}{\frac{7}{x}}\) is equivalent to multiplying by the reciprocal, which results in \(\frac{x}{7}\).

• Look to combine like terms wherever possible to reduce complexity.
• Factor out common elements to simplify the computational process.
• Be alert for opportunities to cancel terms that appear in both the numerator and denominator.

The goal is to rewrite expressions in their simplest, most readable forms, allowing you to clearly see and interpret the behavior of the given functions. By mastering simplification techniques, you enhance your problem-solving toolkit, making math more approachable.