Problem 75 - Algebra 2

Question

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (g \circ f)(x) $$

Step-by-Step Solution

Answer

The simplified version of the composed function \(g \circ f\) is \(g(f(x)) = 2x + 7\).

Step 1: Substitute \(f(x)\) into \(g(x)\)

Replace every \(x\) in the equation for \(g(x)\) with the function \(f(x)=4x\). The composed function \(g \circ f\) is: \(g(f(x)) = \frac{1}{2}(f(x)) + 7\). Substituting \(f(x)\) gives: \(g(f(x)) = \frac{1}{2}(4x) + 7\)

Step 2: Simplify the Equation

Now simplify the equation to find the simplest form. This entails distributing \( \frac{1}{2} \) into \(4x\). The simplified form of \(g(f(x))\) therefore becomes: \(g(f(x)) = 2x + 7\).

Key Concepts

Substitution Method

In the world of mathematics, the substitution method plays a critical role, especially when dealing with functions. It's like swapping one expression with another to transform complex equations. In our exercise, we are composing two functions, namely \( f(x) \) and \( g(x) \). To begin, we substitute \( f(x) = 4x \) into \( g(x) = \frac{1}{2}x + 7 \). This means wherever you see \( x \) in \( g(x) \), you replace it with \( 4x \). This process helps us understand how one quantity influenced by \( f(x) \) can further be influenced by \( g(x) \). This technique is foundational in solving many mathematical problems, as it allows us to express one function in terms of another, thus offering a clearer perspective on the interplay between different variables.

Simplifying Expressions

Once we've substituted the expressions, our next step is simplifying them. Simplifying is like clearing out the clutter from an equation to reveal its simplest form. After substituting \( f(x) \) into \( g(x) \), we have \( g(f(x)) = \frac{1}{2}(4x) + 7 \). The goal here is to make the equation easier to read and work with. For this task, distribute the \( \frac{1}{2} \) to \( 4x \) inside the parentheses:

• Multiply \( \frac{1}{2} \times 4x = 2x \)
• Then add \( 7 \) to the result: \( 2x + 7 \)

Now, the expression \( g(f(x)) = 2x + 7 \) is simple and ready for further interpretation or computation. Simplifying expressions is vital in function composition for clarity and ease of use.

Composite Functions

Composite functions allow us to combine two different functions into one, providing a new perspective on how inputs and outputs are connected. This concept is beautifully illustrated in our exercise with the function notation \( (g \circ f)(x) \), which means "\( g \) composed with \( f \)."To compose functions, you apply the function \( g \) to the result of \( f(x) \). We subjected \( g(x) = \frac{1}{2}x + 7 \) to this process by substituting \( f(x) = 4x \), making \( (g \circ f)(x) = g(f(x)) \).The final simplified expression \( g(f(x)) = 2x + 7 \) is not just any output, but a new function that summarizes the effect of first applying \( f \), then \( g \), to \( x \). In practice, understanding composite functions helps us in breaking down complex function interactions into manageable parts and grasping the sophisticated structures within different systems.