Problem 53 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=x+4, g(x)=2 x+1$$

Step-by-Step Solution

Answer

a) \(f \circ g(x) = 2x + 5\), b) \(g \circ f(x) = 2x + 9\), c) \(f \circ g(2) = 9\), d) \(g \circ f(2) = 13\)

Step 1: Determine (f ∘ g)(x)

To find \(f \circ g(x)\), we substitute g(x) into f(x). Therefore, it becomes \(f(g(x)) = f(2x + 1)\). Substituting the value of f(x) we get \(f(2x + 1) = (2x + 1) + 4 = 2x + 5 \)

Step 2: Determine (g ∘ f)(x)

Now, for \(g \circ f(x)\), we substitute f(x) into g(x). Thus, it simplifies to \( g(f(x)) = g(x + 4) \). While substituting the expression for g(x), we get \(g(x + 4) = 2(x+4) + 1 = 2x + 8 + 1 = 2x + 9 \)

Step 3: Evaluate (f ∘ g)(2)

Using the formula for \(f \circ g(x)\) obtained in step 1, we set x = 2 into that upfront: \(f \circ g(2) = 2(2) + 5 = 4 + 5 = 9 \)

Step 4: Evaluate (g ∘ f)(2)

We plug x = 2 to the \(g \circ f(x)\) equation obtained before: \(g \circ f(2) = 2(2) + 9 = 4 + 9 = 13 \)

Key Concepts

Composite Functions

Understanding composite functions is a crucial aspect of precalculus mathematics. A composite function is created when one function is applied immediately after another function. The notation \(f \circ g)(x)\) represents such a composition, where \(g\) is applied first to \(x\), and then \(f\) is applied to the result of \(g(x)\).

For example, given the functions \(f(x) = x + 4\) and \(g(x) = 2x + 1\), we find the composite function \(f \circ g\) by applying \(g\) to \(x\) first, and then applying \(f\) to \(g(x)\), giving us \(f(g(x)) = f(2x + 1)\). When we substitute the expression for \(f(x)\) into this equation, it simplifies to \(2x + 5\), which is the composite function \(f \circ g)(x)\).

It's important not to confuse this with \(g \circ f\) which reverses the order of application. The distinct results of composite functions highlight the principle that the order of function composition matters.

Function Evaluation

Function evaluation involves substituting a specific value for the variable in the function's expression. In the context of composite functions, evaluating \(f \circ g)(2)\) involves two steps. First, we find the expression for \(f \circ g)(x)\) as previously mentioned. Then, we substitute \(x = 2\) in the composite function's formula to calculate the specific output. In this case, \(f \circ g)(2) = 2(2) + 5 = 9\).

This is a fundamental concept in mathematics as it allows us to determine the output of a function for a particular input, which is essential for graphing functions, solving equations, and understanding the behavior of mathematical models in various applications.

Precalculus Mathematics

Precalculus mathematics serves as the foundation to calculus, covering a variety of topics including functions, equations, inequalities, sequences, and series. It's designed to prepare students with the skills and knowledge required to understand and tackle calculus problems.

Composite function evaluation is integral to precalculus. Students often work with exercises like our example \(f(x) = x + 4\) and \(g(x) = 2x + 1\), to practice the manipulation and evaluation of functions which are essential skills. The ability to comprehend and apply these principles paves the way for future studies in higher-level mathematics and its applications in science, engineering, economics, and other fields.