Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=3 x, \quad g(x)=2 x+5\)

Step-by-Step Solution

Answer

The function compositions are \(f \circ g(x) = 6x + 15\), \(g \circ f(x) = 6x + 5\), and \(f \circ f(x) = 9x\).

Step 1: Find \(f \circ g\)

To find \(f \circ g\), plug into \(f(x)\) the equation for \(g(x)\). In this case, replace \(x\) in \(f(x) = 3x\) with \(g(x) =2x+5\). So \(f \circ g(x) = 3(2x+5) = 6x + 15\).

Step 2: Find \(g \circ f\)

To find \(g \circ f\), plug into \(g(x)\) the equation for \(f(x)\). Replacement of \(x\) in \(g(x) = 2x+5\) with \(f(x) = 3x\). So \(g \circ f(x) = 2(3x) + 5 = 6x+5\).

Step 3: Find \(f \circ f\)

To find \(f \circ f\), plug into \(f(x)\) the equation for \(f(x)\). It will be replacement of \(x\) in \(f(x) = 3x\) with \(f(x) = 3x\). So \(f \circ f(x) = 3(3x) = 9x\).

Key Concepts

Function Operations

Understanding function operations is vital whenever we're working with algebraic expressions. Fundamentally, a function operation allows us to combine two functions using addition, subtraction, multiplication, or division. But we're not limited to these basic operations; we can also perform operations that involve plugging the output of one function directly into another function, which brings us to the concept of function composition.

This process is akin to a relay race: the output of the first function is passed as the input to the second function, creating a chain of operations that produces a new function. When working with function operations, it's crucial to pay attention to the order of operations, as the composition of functions is not commutative - this means that switching the order of the functions typically yields different results. For student comprehension, illustrating the process using numerical examples can greatly aid in the understanding of the abstract algebraic process.

Algebraic Functions

An algebraic function is essentially a mathematical expression that combines numbers and at least one variable using the operations of addition, subtraction, multiplication, division, and taking roots. This can be something as simple as a linear function, like the one described with \( f(x) = 3x \) in our exercise, or more complex functions involving polynomials or rational expressions.

For students to grasp algebraic functions, familiarity with various forms and their corresponding graphs is beneficial. For instance, recognizing that the equation \( g(x) = 2x+5 \) represents a line can help students predict the shape of the graph and the behavior of the function. When delving deeper into algebra, understanding how these functions work together, how they are transformed, and their inversions becomes essential. Breakdowns of algebraic functions into their individual components help students see the underlying structure and reinforce their skills in manipulating these expressions algebraically.

Composite Functions

Composite functions are at the heart of our given exercise. A composite function is the result of the composition of two functions; this is like putting one function inside another. The notation \( f \circ g \) represents the composite function where \( g \) is applied first, and then \( f \) is applied to the result of \( g \) - the output of \( g(x) \) becomes the input of \( f(x) \).

It's crucial for students to follow the correct order in composition, as reversing the order can produce an entirely different result. In the exercise, applying \( f \circ g \) is not the same as \( g \circ f \) because the functions \( f(x) \) and \( g(x) \) are distinct and have different algebraic forms. Moreover, the concept of \( f \circ f \) introduces students to the idea of a function composed with itself, which can lead to new insights regarding the behavior of functions over iterations. Visual aids or graphing calculators can be extremely beneficial when demonstrating how these compositions affect graphs and function behaviors.