Problem 67 - A Graphical Approach to College Algebra

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=5, \quad g(x)=x$$

Step-by-Step Solution

Answer

Each composition is 5, with domain \((-\infty, \infty)\).

Step 1: Understand Functions

The functions provided are simple. The function \(f(x) = 5\) is a constant function, meaning it outputs 5 for any value of \(x\). The function \(g(x) = x\) is an identity function, meaning it outputs the input value itself.

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

The notation \((f \circ g)(x)\) means we substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = f(x)\). Since \(f(x) = 5\), \(f(g(x)) = f(x) = 5\). Thus, \((f \circ g)(x) = 5\). The domain of a constant function is all real numbers, \((-\infty, \infty)\).

Step 3: Compute \((g \circ f)(x)\)

The notation \((g \circ f)(x)\) means we substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(5) = 5\) since \(g(x) = x\) and therefore returns the input. Thus, \((g \circ f)(x) = 5\). The domain of this constant function is again all real numbers, \((-\infty, \infty)\).

Step 4: Compute \((f \circ f)(x)\)

The notation \((f \circ f)(x)\) means we substitute \(f(x)\) into \(f(x)\): \(f(f(x)) = f(5) = 5\). Thus, \((f \circ f)(x) = 5\). Like the previous functions, the domain is all real numbers, \((-\infty, \infty)\).

Step 5: Summarize Findings

For each composition \((f \circ g)(x)\), \((g \circ f)(x)\), and \((f \circ f)(x)\), the result is the constant function 5. The domain for each composition is all real numbers because a constant function does not place any restriction on the domain.

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is defined as a function that always returns the same constant value no matter what input it receives. For example, the function \(f(x) = 5\) is a constant function because it outputs the number 5 for any value of \(x\).

Constant functions are straightforward and predictable. They have a few properties that make them unique:

• The graph of a constant function is a horizontal line on the Cartesian plane.
• Its slope is zero, indicating no change in the output as the input changes.

Because the output is always the same, constant functions can often simplify more complex compositions.

• This makes them useful in mathematical models where a steady state or fixed value is desired.

Identity Function

Identity functions represent another basic concept in mathematics. An identity function is a type of function that returns its input without any modification. In mathematical terms, the function \(g(x) = x\) is an identity function. This means if you input 7 into the function, it will output 7. Likewise, input -3.5 and get -3.5 as output.

Identity functions are characterized by these properties:

• Their graph is a straight line that passes through the origin with a 45-degree angle against both the x and y axes.
• They serve as a neutral element in the system of functions, much like the number 0 in addition or 1 in multiplication.

They help in maintaining the original properties of inputs, and they're essential in operations that involve manipulating functions, like composition.

In the problem's context, substituting an identity function within another function doesn’t alter the result.

Domain of a Function

The domain of a function is critical to understanding the function's behavior. It defines all the possible input values \(x\) for which the function is defined. In other words, the domain is the complete set of valid inputs that won't result in mathematical errors, like dividing by zero or taking the square root of a negative number (in the context of real numbers).

For constant functions like \(f(x) = 5\), the domain is very simple: it's all real numbers, written as \(( -\infty , \infty )\).

• This is because no real number input will affect the output; it will always be 5.
• There are no restrictions such as division by zero or undefined results.

With identity functions, the domain is also all real numbers. Since \(g(x) = x\), any real number entered remains unchanged and is perfectly valid as output.

Understanding domains allows students to determine where a function is applicable and meaningful, both practically and theoretically.