Problem 46 - College Algebra

Question

Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=\frac{1}{4 x+5}$$

Step-by-Step Solution

Answer

The functions f and g that compose h when applied in a chain can be defined as \(f(x) = \frac{1}{x}\) and \(g(x) = 4x + 5\). Thus, the function \(h(x)\) can be represented as \(h(x) = (f\circ g)(x)\)

Step 1: Select the Function g

Begin by taking a look at the mathematical operation involving x in the function h. In this case, the basic operation is multiplication and addition done on x to yield \(4x + 5\). This suggests that g(x) could be \(4x + 5\).

Step 2: Select the Function f

Next, consider what happens to the result after obtaining \(4x + 5\). In the given function h, the result is taken as the denominator in a fraction where the numerator is 1, basically the operation of taking a reciprocal. Therefore, let's choose the function f to handle this operation by setting \(f(x) = \frac{1}{x}\).

Step 3: Combining the Functions

Now, to create the composition of the functions, we substitute g(x) into f to get \(f(g(x)) = f(4x + 5) = \frac{1}{4x + 5}\)

Key Concepts

Algebraic Functions

In mathematics, an algebraic function represents a relationship where each input value is associated with exactly one output value, typically described using polynomial equations. They can take various forms and include operations such as addition, subtraction, multiplication, division, and the use of exponents.

Understanding algebraic functions is crucial as they are the building blocks of more complex mathematical concepts. When it refers to the function composition, like in our exercise example, \(h(x)=\frac{1}{4x+5}\), we're looking at two functions \(f\) and \(g\) that combine to create a new function \(h\) through a specific algebraic relationship. In the given problem, \(g\) is based on a linear function \(4x + 5\), and \(f\) is a function that takes the reciprocal of any input. The combination of these two forms the functional composition defining \(h\).

Mathematical Operations

Mathematical operations are the foundational procedures in arithmetic and algebra, including addition, subtraction, multiplication, division, and more advanced operations such as taking roots and reciprocals. These operations form the basis upon which algebraic functions are constructed and understood.

In function composition, the ordering of these operations is essential to derive the correct formula. Consider our example where \(h(x)\) is broken down into steps involving simpler functions. The initial step involves linear operations on \(x\) (multiplication by 4 and addition of 5), thus establishing a foundation for further manipulation through higher-level operations like taking a reciprocal, which falls into the purview of function \(f\). By precisely defining and arranging these operations, we are able to compose functions in a way that they accurately represent complex relationships in algebra.

Reciprocal Function

The reciprocal function, simply put, is a function that assigns to an input \(x\) the output \(1/x\). This function is critical in various fields of mathematics because it embodies the operation of inversion with respect to multiplication. A reciprocal can transform a division problem into a multiplication problem, often simplifying the computation.

In our exercise, the function \(f\) is designed to take the reciprocal of its input, effectively flipping a number over the value of one. This transformation is central to creating the composite function \(h(x)\) from the earlier identified functions \(f\) and \(g\). The reciprocal function's action on \(4x + 5\) gives rise to the composite function \(h(x)\), which showcases how a complicated expression can result from the application of very basic mathematical operations when they are combined in a functional composition.