Problem 17 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=2 x, g(x)=x+7$$

Step-by-Step Solution

Answer

The results are (f • g)(x) = 2x + 14, (g • f)(x) = 2x + 7, and (f • g)(2) = 18.

Step 1: Find (f • g)(x)

This involves applying function \(f\) to the result of function \(g\) on \(x\). Function \(g(x) = x + 7\), substitute this into \(f(x)\) to give: \[f(g(x)) = 2(g(x)) = 2(x+7) = 2x +14\].

Step 2: Find (g • f)(x)

This involves applying function \(g\) to the result of function \(f\) on \(x\). Function \(f(x) = 2x\), substitute this into \(g(x)\) to give: \[g(f(x)) = (2x) + 7 = 2x + 7\].

Step 3: Find (f • g)(2)

This involves taking the result of the first composition, \((f \circ g)(x) = 2x + 14\), and substituting \(x = 2\) to give: \[(f \circ g)(2) = 2(2) + 14 = 4 + 14 = 18\].

Key Concepts

Function Operations

When we talk about function operations, we mean applying one function to the results of another function. This is called composition. It's like feeding the output of one function directly into another. In the given exercise, we have two functions, \( f(x) = 2x \) and \( g(x) = x + 7 \).

• In part a, we compose \( f \) and \( g \) as \( (f \circ g)(x) \), which means we substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = 2(g(x)) = 2(x+7) = 2x + 14 \).
• In part b, we reverse the order in \( (g \circ f)(x) \), substituting \( f(x) \) into \( g(x) \): \( g(f(x)) = (2x) + 7 = 2x + 7 \).
• In part c, we find \( (f \circ g)(2) \) by substituting \( x = 2 \) into \( (f \circ g)(x) = 2x + 14 \), giving \( 4 + 14 = 18 \).

Function composition is essential in mathematics. It helps us understand how complex processes can be broken down into simpler steps, similar to processing information through different stages.

Algebraic Expressions

Algebraic expressions are crucial in composing functions. An algebraic expression is a combination of numbers, variables, and operations. When we substitute one function into another, we're essentially dealing with algebraic expressions. Let's think about it practically.

• In our problem, \( f(x) = 2x \) is an algebraic expression where \( x \) is multiplied by 2.
• Meanwhile, \( g(x) = x + 7 \) means 7 is added to \( x \).
• When we perform these substitutions, like \( f(g(x)) = 2(x + 7) \), we must respect order of operations: parentheses first, then multiplication.

Understanding algebraic expressions helps simplify complex problems. We see how individual components interact and affect each other, paving the way for solving and interpreting math problems efficiently.

College Algebra

College algebra covers many topics, including functions and their compositions. At this level, students need to master operations like addition, subtraction, multiplication, and division of functions, as well as composition. It expands your mathematical understanding and problem-solving skills.

• This exercise is an example of what you might encounter in college algebra. Here, students are expected to handle compositions like \( (f \circ g)(x) \) and \( (g \circ f)(x) \).
• College algebra teaches how to apply these compositions in real-world scenarios, involving functions that model real-life situations.
• It's also crucial for learning more advanced topics in calculus and beyond, as understanding function behavior is foundational.

Achieving proficiency in college algebra is important. It builds a strong groundwork for future studies and everyday problem-solving, ensuring you’re prepared for the diverse challenges of advanced mathematics.