Problem 64 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=6 x-3, g(x)=\frac{x+3}{6}$$

Step-by-Step Solution

Answer

The resulting functions and values are: (a) \(f \circ g(x) = x\), (b) \(g \circ f(x) = x\), (c) \(f \circ g(2) = 2\), and (d) \(g \circ f(2) = 2\).

Step 1: Calculate \(f \circ g(x)\)

The composition of functions \(f \circ g(x)\) means applying \(g(x)\) first and then \(f(x)\) to the result of \(g(x)\). So, \(f(g(x))= f(\frac{x+3}{6})= 6 * (\frac{x+3}{6}) - 3 = x + 3 - 3 = x.

Step 2: Calculate \(g \circ f(x)\)

The composition of functions \(g \circ f(x)\) means applying \(f(x)\) first and then \(g(x)\) to the result of \(f(x)\). So, \(g(f(x)) = g(6x-3) = \frac{6x-3+3}{6} = \frac{6x}{6} =x.

Step 3: Calculate \(f \circ g(2)\)

Plugging \(2\) into the composite function \(f \circ g(x)\) gives \(f(g(2)) = f(\frac{2+3}{6}) = f(\frac{5}{6}) = 6*\frac{5}{6}-3 = 5-3 = 2.

Step 4: Calculate \(g \circ f(2)\)

Plugging \(2\) into the composite function \(g \circ f(x)\) gives \(g(f(2)) = g(6*2-3) = g(9) = \frac{9+3}{6}=\frac{12}{6}=2.

Key Concepts

Composite Function

A composite function is the result of applying one function to the result of another function. It's like stacking the functions where the output from one function becomes the input to the next. This process can be likened to a factory assembly line where one machine takes the product from the previous machine and performs its own operation on it.

Understanding composite functions is crucial in mathematics as it allows you to create new functions by combining existing ones. The notation \(f \circ g\)(x) denotes the composite function, where \(g\) is applied first, and then \(f\) is applied to the result of \(g(x)\). In our example, \(f(x) = 6x - 3\) is composed with \(g(x) = \frac{x+3}{6}\), resulting in the composite functions \(f \circ g\)(x) and \(g \circ f\)(x).

Function Operations

Performing operations with functions is similar to arithmetic operations with numbers, but instead of numbers, you're working with entire expressions. The basic operation we are focusing on here is function composition, where two functions are combined to form a single function.

It's important to understand that the order in which you compose functions matters. \(f \circ g\) is generally different from \(g \circ f\). For instance, if \(f(x)\) represents doubling a number and \(g(x)\) represents adding five, then \(f \circ g\)(x) would represent doubling a number after adding five, while \(g \circ f\)(x) would involve adding five after doubling the number. As deduced from our exercise, though, there are cases where \(f \circ g\)(x) and \(g \circ f\)(x) could be the same, which is an interesting property known as the functions being commutative under composition.

Function Composition Steps

The process of creating a composite function involves specific steps that ensure the functions are combined accurately. Our working example demonstrates a clear methodology to follow:

Step 1: Understand the Order

The first step is to identify which function is \(g\) and which is \(f\), and in which order you need to apply them. This is deciphered from the notation, remembering that \(g\) is used first in \(f \circ g\)(x).

Step 2: Substitute and Simplify

Next, you substitute \(g(x)\) into \(f(x)\), replacing every occurrence of 'x' in \(f\) with \(g(x)\). After the substitution, simplify the result to find the expression that represents \(f \circ g\)(x).

Step 3: Evaluate

If needed, like in parts (c) and (d) of our example, plug in a specific value for 'x' into your composite function and simplify to arrive at the number that represents the composite function's output for that input.

The clear depiction of these steps helps students to systematically handle function composition, a foundational concept in algebra and higher-level mathematics.