Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find \((\mathbf{a})\) \(\boldsymbol{f} \circ \boldsymbol{g},(\mathbf{b}) \boldsymbol{g} \circ \boldsymbol{f},\) and, if possible, \((\mathbf{c})(\boldsymbol{f} \circ \boldsymbol{g})(\mathbf{0}).\) $$f(x)=x^{3}, \quad g(x)=\frac{1}{x}$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = \frac{1}{x^{3}},\ (g \circ f)(x) = \frac{1}{x^{3}}.\ (f \circ g)(0)\ does not exist as it results in division by zero.

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

The composition \(f \circ g\) means we substitute \(g(x)\) into \(f(x)\). Thus, \(f(g(x)) = \left(\frac{1}{x}\right)^{3} = \frac{1}{x^{3}}\).

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

Similarly, the composition \(g \circ f\) means we substitute \(f(x)\) into \(g(x)\). Thus, \(g(f(x)) = \frac{1}{(x^{3})} = \frac{1}{x^{3}}.\) Note that in this case the compositions \(f \circ g\) and \(g \circ f\) are the same, but this is not generally true.

Step 3: Evaluate \((f \circ g)(0)\)

We are asked to find the value of \((f \circ g)(0)\), this requires substituting 0 into \((f \circ g)(x) = \frac{1}{x^{3}}\). But this is not defined as we cannot divide by zero, so \((f \circ g)(0)\) does not exist.

Key Concepts

Composite Function

Understanding composite functions is critical for students as it is a foundational concept in algebra and precalculus. A composite function can be thought of as a function within a function. It involves applying one function to the results of another. To find the composite function \( f \circ g \), you perform the function \( g \) and then use the output of \( g \) as the input for the function \( f \).

In the given exercise, the function \( g(x) \) is first applied, which gives us \( 1/x \) and then, this result is put through the function \( f(x) \), resulting in \( (1/x)^3 \).

Function Operation

A function operation like the composition of functions is a process where you combine two functions using a specific operation. This could involve addition, subtraction, multiplication, division, or composition. The operation of composition is unique because it isn't about performing arithmetic on the functions directly but rather applying the entire function to another function's output.

It is important to note that the order in which functions are composed matters. The exercise illustrates this by asking for \( f \circ g \) and \( g \circ f \) separately. While in this instance both result in the same expression, \( \frac{1}{x^3} \), usually, \( f \circ g \) will not equal \( g \circ f \). Always pay close attention to the order of operations in function composition.

Undefined Value

In mathematics, an undefined value occurs when an expression doesn't have a meaningful result. This typically happens with operations that are not permissible, like division by zero. In our exercise, the composite function \( (f \circ g)(0) \) is undefined because it involves dividing one by zero, which is not allowed.

This concept is crucial to grasp because it highlights limitations within a function's domain—the set of all possible inputs. For continuous learning and correct solution of problems, recognizing when a function will be undefined prevents missteps and paves the way for a better understanding of function behavior.