Problem 66 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=\frac{x-3}{2}, g(x)=2 x-3 $$

Step-by-Step Solution

Answer

So, \(f(g(x)) = x - 3\) and \(g(f(x)) = x - 6\).

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

First, replace all instances of \(x\) in the function \(f(x)\) with the function \(g(x)\): \[f(g(x)) = f(2x - 3) = \frac{2x - 3 - 3}{2}.\] Then, simplify \[f(g(x)) = \frac{2x - 6}{2} = x - 3.\]

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

Now, replace all instances of \(x\) in the function \(g(x)\) with \(f(x)\):\[g(f(x)) = g\left(\frac{x-3}{2}\right) = 2 \cdot \frac{x-3}{2} - 3.\] Then, simplify \[g(f(x)) = x - 3 - 3 = x - 6.\]

Key Concepts

Composite Functions

When working with composite functions, you are essentially combining two functions into one new function. This process involves taking the output of one function and using it as the input of another. Think of composite functions like a conveyor belt: the first function processes an input and passes its result to the second function. To find the composite function \( f(g(x)) \), we substitute the function \( g(x) \) into \( f(x) \). Similarly, to find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). This method is crucial in various mathematical applications because it allows us to study the behavior of nested relationships between functions. Understanding how composite functions work is essential for more advanced mathematics, including calculus and algebra.

Simplification

Simplification is about making an expression more accessible and easy to interpret. It involves reducing complex formulas or problems to their simplest forms. In the case of composite functions, after substituting one function into another, you'll often find an expression that needs simplification. For \( f(g(x)) = \frac{2x - 6}{2} \), the simplification process involves dividing both terms in the numerator by 2, resulting in \( x - 3 \). Simplifying often makes the mathematical relationships more apparent and can lead to easier problem-solving. This step is essential to clear the clutter and focus on the core value of the function or expression.

Functions Pair

Function pairs refer to a set of two functions that are used together to form composite functions. In this exercise, we're dealing with \( f(x) = \frac{x-3}{2} \) and \( g(x) = 2x-3 \), which we evaluated both ways: \( f(g(x)) \) and \( g(f(x)) \). Examining these pairs helps highlight relationships and interactions between the two functions.
This process allows us to explore how changes in one function might affect the other, which is foundational knowledge for more complex problem solving, including dynamic systems and iterative processes. Understanding function pairs enables students to develop a deeper comprehension of how combined functions behave.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify algebraic expressions. It's a fundamental skill in solving equations and understanding functions. When you substitute one function into another, as in \( g(f(x)) = 2 \cdot \frac{x-3}{2} - 3 \), you utilize algebraic manipulation to simplify it to \( x - 6 \). This involves distributing the constant 2 across the expression and then subtracting 3.

• Distributing constants or factors.
• Combining like terms.
• Factoring expressions.

By practicing these skills, you build a toolkit that makes complex algebraic problems more manageable and solutions more straightforward to achieve. Algebraic manipulation is integral in mathematics as it ensures clarity and precision in expressing relationships and finding solutions.