Problem 123 - College Algebra with Modeling and Visualization

Question

Let \(f(x)=k\) and \(g(x)=a x+b,\) where \(k, a,\) and \(b\) are constants. (a) Find \((f \circ g)(x) .\) What type of function is \(f \circ g ?\) (b) Find \((g \circ f)(x) .\) What type of function is \(g \circ f ?\)

Step-by-Step Solution

Answer

(f \circ g)(x) is a constant function k; (g \circ f)(x) is a constant function ak + b.

Step 1: Understand Function Composition

Function composition involves applying one function to the result of another function. The notation \((f \circ g)(x)\) means that we first apply \(g(x)\) and then apply \(f\). Similarly, \((g \circ f)(x)\) means we first apply \(f(x)\) and then \(g\).

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

To find \((f \circ g)(x)\), we substitute \(g(x)\) into \(f(x)\). Since \(f(x) = k\), \(f\) is a constant function:\[ (f \circ g)(x) = f(g(x)) = f(a x + b) = k. \]This means \((f \circ g)(x)\) is simply the constant \(k\) for all \(x\).

Step 3: Identify the Type of \(f \circ g(x)\)

Since \((f \circ g)(x) = k\), the composition results in a constant function.

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

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). Since \(f(x) = k\), we have:\[ (g \circ f)(x) = g(f(x)) = g(k) = a(k) + b. \]This simplifies to the constant \(ak + b\).

Step 5: Identify the Type of \(g \circ f(x)\)

\((g \circ f)(x) = ak + b\) is also a constant function, as it does not depend on \(x\).

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is defined by its equation, typically in the form of \( f(x) = k \), where \( k \) is a constant value. This means that no matter what input \( x \) you provide, the output will always be the same value \( k \). Constant functions are flat horizontal lines when graphed on the Cartesian coordinate system.

Here are some key features of constant functions:

• The graph of a constant function is parallel to the x-axis.
• It does not have any changes; hence, its derivative is zero.
• It is both even and continuous everywhere.

Understanding constant functions helps lay the groundwork for more complex functions, as they often appear in function composition scenarios. When you compose a constant function with any other function, the result is usually another constant function, as demonstrated in the original exercise above.

Linear Function

Linear functions are fundamental in algebra and calculus. They are defined by the equation \( g(x) = ax + b \), where \( a \) and \( b \) are constants. The variable \( x \) is raised only to the first power, making it a linear equation.

Some important characteristics of linear functions include:

• The graph of a linear function is a straight line.
• The slope of the line is determined by \( a \), which affects the steepness and direction.
• The y-intercept, where the line crosses the y-axis, is determined by \( b \).

When you are working with a linear function in the context of function composition, such as combining it with another function like a constant function, the properties of the linear function influence the result. However, if you compose a linear function with a constant function, such as in the problem \((g \circ f)(x)\), you'll notice that the x-dependence of the linear function is lost, leaving a constant value that does not change with \( x \).

Algebraic Function

An algebraic function is a broad classification of functions that are expressed using algebraic expressions. This includes operations such as addition, subtraction, multiplication, division, and raising to a power. Both constant and linear functions are types of algebraic functions.

Key points about algebraic functions:

• They can be simple, like constants or linear functions, or more complex combinations of various terms.
• They follow the conventional rules of algebra.
• They are everywhere continuous within their domain.

In the context of the original exercise, the function compositions \((f \circ g)(x)\) and \((g \circ f)(x)\) are examples of how algebraic properties are used and further simplify functions. This exercise involves composing a constant function with a linear function, resulting in either constant functions, illustrating the versatile and simplifying nature of algebraic functions when composed.