Problem 57 - Brief Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=2 x-6 ; g(x)=\frac{x}{2}+3 $$

Step-by-Step Solution

Answer

a. \(f(g(x)) = x\), b. \(g(f(x)) = x\)

Step 1: Evaluate f(g(x))

First, substitute the expression for \(g(x)\) into \(f(x)\). Given \(g(x) = \frac{x}{2} + 3\), substitute into \(f(x) = 2x - 6\):\[ f(g(x)) = 2\left(\frac{x}{2} + 3\right) - 6. \]

Step 2: Simplify the Expression for f(g(x))

Distribute the 2 into the terms in the parentheses:\[ f(g(x)) = 2 \cdot \frac{x}{2} + 2 \cdot 3 - 6. \]Simplifying gives:\[ f(g(x)) = x + 6 - 6 = x. \]

Step 3: Evaluate g(f(x))

Now, substitute the expression for \(f(x)\) into \(g(x)\). Given \(f(x) = 2x - 6\), substitute into \(g(x) = \frac{x}{2} + 3\):\[ g(f(x)) = \frac{2x - 6}{2} + 3. \]

Step 4: Simplify the Expression for g(f(x))

First, simplify \(\frac{2x - 6}{2}\):\[ \frac{2x - 6}{2} = \frac{2x}{2} - \frac{6}{2} = x - 3. \]Now, substitute back into \(g(f(x))\):\[ g(f(x)) = x - 3 + 3. \]Simplifying gives:\[ g(f(x)) = x. \]

Key Concepts

Composite Functions

In mathematics, composite functions involve combining two or more functions to create a new function. Imagine you have two functions, \(f(x)\) and \(g(x)\). When you find the composite function \(f(g(x))\), you are basically plugging \(g(x)\) into \(f(x)\). This means, wherever you see \(x\) in \(f(x)\), you replace it with \(g(x)\). In our exercise, \(f(x) = 2x - 6\) and \(g(x) = \frac{x}{2} + 3\). To find \(f(g(x))\), you take the expression for \(g(x)\) and insert it into \(f(x)\) like this:

• Start with the expression for \(g(x) = \frac{x}{2} + 3\).
• Substitute it into \(f(x)\) to get \(f(g(x)) = 2\left(\frac{x}{2} + 3\right) - 6\).

The goal is to create a new function that follows the path of both \(f(x)\) and \(g(x)\). It's like a chain of operations! This can be done in the reverse as well with \(g(f(x))\). Employing composite functions allows us to understand and visualize how outputs of one function become inputs for another.

Simplification

Simplification is about reducing mathematical expressions to their simplest form. This is important because it makes it easier to understand and work with equations. Let's examine how it applies to our composite functions. For \(f(g(x))\), after substituting \(g(x)\) into \(f(x)\), the expression becomes \(2\left(\frac{x}{2} + 3\right) - 6\). Through simplification:

• First, distribute the 2: \(2 \cdot \frac{x}{2} + 2 \cdot 3\).
• This leads to \(x + 6\).
• Then, combine terms to get \(x + 6 - 6 = x\).

Simplifying makes the entire function clearer and shows that \(f(g(x)) = x\).Similarly, for \(g(f(x))\):

• Substitute \(f(x)\) into \(g(x)\) to get \(\frac{2x - 6}{2} + 3\).
• Simplify to \(x - 3 + 3\).
• Combine terms to conclude \(g(f(x)) = x\).

Simplification helps in recognizing patterns and ensuring that functions are in their most usable form.

Substitution Method

The substitution method is a helpful technique used in forming and solving composite functions. Essentially, it's about replacing a variable with another expression so you can simplify or solve a function. Substitution acts as if you are swapping ingredients in a recipe but maintaining the end dish! Let's see how it is used in our example.For \(f(g(x))\):

• You pick the expression of \(g(x) = \frac{x}{2} + 3\).
• Insert this into \(f(x)\) wherever there is an \(x\), resulting in: \(2\left(\frac{x}{2} + 3\right) - 6\).
• This use of substitution transforms the original problem into something more manageable.

Employing substitution in \(g(f(x))\):

• Begin with \(f(x) = 2x - 6\) and place it inside the \(g(x)\).
• So \(g(f(x)) = \frac{2x - 6}{2} + 3\).

This technique simplifies not just the numbers but also the process. Learning this method equips you with a powerful tool in your mathematical toolbelt, allowing seamless transition between functions and revealing elegant patterns.