Problem 64 - College Algebra

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

\(f(g(x)) = x, g(f(x)) = x, f(g(2)) = 2, g(f(2)) = 2\)

Step 1: Find (f∘g)(x)

Replace g(x) in f(x). This gives \(f(g(x)) = 6 * \frac{x+3}{6} - 3 = x + 3 - 3 = x\)

Step 2: Find (g∘f)(x)

Replace f(x) in g(x). This gives \(g(f(x))= \frac{6x - 3 + 3}{6} = x\)

Step 3: Calculate (f∘g)(2)

Replace \(x\) with \(2\) in \(f(g(x))\). This gives \(f(g(2)) = 2\)

Step 4: Calculate (g∘f)(2)

Replace \(x\) with \(2\) in \(g(f(x))\). This gives \(g(f(2)) = 2\)

Key Concepts

Function Notation

Function notation is a systematic method of expressing the dependence of one quantity on another. It is often used in mathematics to define functions in a concise way. Instead of saying "the function of x is 2x + 3," you use function notation and write it as "\(f(x) = 2x + 3\)." This notation clearly shows the function \(f\) and the variable \(x\) it acts upon.

• The letter, such as \(f\), identifies the function, while \(x\) represents the variable.
• Function notation allows for easy substitution of different variables. For instance, you can find \(f(2)\) by replacing \(x\) with 2 in the equation \(f(x) = 2x + 3\).

Using function notation, you can easily describe complex relationships and plug in values or other functions to see how they behave. Understanding function notation is fundamental, especially when dealing with composite functions.

Composite Functions

Composite functions involve combining two functions so that the output of one function becomes the input of another. This is often denoted by \((f \circ g)(x)\) or \(f(g(x))\). It reads as "\(f\) composed with \(g\)," and provides a powerful way to create a new function from existing ones.

• To find a composite function such as \((f \circ g)(x)\), you substitute the entire function \(g(x)\) into the function \(f(x)\).
• This can be thought of as a function "inside" another function, where the result of \(g(x)\) directly influences\( f(x)\).

In practical terms, suppose \(f(x) = 6x - 3\) and \(g(x) = \frac{x+3}{6}\). To determine \((f \circ g)(x)\), you substitute \(g(x)\) into \(f(x)\), thus computing \(f(g(x)) = 6 \cdot \frac{x+3}{6} - 3 = x\). Salesforce of functions allows for streamlined and efficient calculations and transformations.

Evaluation of Functions

Evaluation of functions involves substituting a specific value into the function to obtain a numerical result. This is crucial for understanding how functions behave for given inputs. For example, evaluating \(f(x)\) at \(x = 2\) means computing \(f(2)\).

• To evaluate a composite function, first compute the inner function with the given input, followed by the outer function using the result.
• Evaluations of composite functions may sometimes yield the same results as the individual functions, due to specific properties or simplifications.

Using the previous example functions, evaluating \((f \circ g)(2)\) starts with \(g(2)\) which is \(\frac{2+3}{6}\), and the result is plugged into \(f(x)\), yielding \(f(g(2)) = 2\). Similarly, \((g \circ f)(2)\) is evaluated as \(g(f(2))\). This step-by-step approach is fundamental for tackling function-based problems effectively.