Problem 42 - Elementary and Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. $$ (g \circ f)\left(\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

(g \circ f)\left(\frac{1}{3}\right) = \frac{16}{9}.

Step 1: Understand Composite Functions

The symbol \((g \circ f)(x)\) denotes a composite function where function \(f\) is applied first and then function \(g\) is applied to the result of \(f\). Thus, \((g \circ f)(x) = g(f(x))\). We need to apply this process to \(x = \frac{1}{3}\).

Step 2: Evaluate the Inner Function \(f(x)\)

Substitute \(x = \frac{1}{3}\) into the function \(f(x) = 2x + 1\):\[ f\left(\frac{1}{3}\right) = 2 \times \frac{1}{3} + 1 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]So, \(f\left(\frac{1}{3}\right) = \frac{5}{3}\).

Step 3: Evaluate the Outer Function \(g(x)\)

Now apply the function \(g(x) = x^2 - 1\) to the result of the inner function, \(f\left(\frac{1}{3}\right) = \frac{5}{3}\):\[ g\left(\frac{5}{3}\right) = \left(\frac{5}{3}\right)^2 - 1 = \frac{25}{9} - 1 = \frac{25}{9} - \frac{9}{9} = \frac{16}{9} \]Thus, \(g\left(\frac{5}{3}\right) = \frac{16}{9}\).

Step 4: Conclude with the Composite Function

We have evaluated the expression \((g \circ f)\left(\frac{1}{3}\right) = g(f(\frac{1}{3}))\). From step 3, we found this value to be \(\frac{16}{9}\). Therefore, \((g \circ f)\left(\frac{1}{3}\right) = \frac{16}{9}\).

Key Concepts

Function Composition

Composite functions involve performing one function and then applying another function on the result. Imagine each function like a machine: you insert something in, the machine does its job, and out comes a result, which then goes into the next machine. The notation \(g \circ f\)(x) signifies that you first apply the function \(f\) to \(x\) and then apply the function \(g\) to the result of \(f\). So, it looks like making a sandwich: the inner layer (\(f\)) is tackled first and then wrapped with the outer layer (\(g\)).
In mathematical terms, this becomes \((g \circ f)(x) = g(f(x))\). First, compute \((f(x))\), and then take that output as the input for \((g(x))\). Understanding this concept is key to navigating and solving composition problems more easily.

• Step 1: Compute the first function \(f\) using the input.
• Step 2: Use the output from \(f\) as the input for function \(g\).
• Step 3: The result is the output of the composite function.

Remember that keeping track of the order is critical. The result from \(f\) must go into \(g\), not the other way around.

Evaluating Functions

Evaluating a function means finding the output for a given input. Think of this process as turning a crank on a machine to produce a number. Each function has a specific rule that tells you what to do with your input.
Let's take the inner function \(f(x) = 2x + 1\) for example: by substituting \(x = \frac{1}{3}\) directly, you apply the rule of \(f\), which is multiply by 2 and then add 1. This gives you the first output:
\[ f\left(\frac{1}{3}\right) = 2 \times \frac{1}{3} + 1 = \frac{5}{3} \]
The next step involves taking the output \(\frac{5}{3}\) from \(f\) and substituting it into the outer function \(g(x) = x^2 - 1\):
\[ g\left(\frac{5}{3}\right) = \left(\frac{5}{3}\right)^2 - 1 = \frac{16}{9} \]
This two-step approach ensures you correctly evaluate the composite function by working through each function's specific operations.

Function Operations

In mathematics, function operations involve combining or manipulating functions in different ways, which can include addition, subtraction, multiplication, division, and composition, like the one we analyzed here.
Function composition is just one type of operation. Each operation follows its own rules and definitions, which dictate how functions interact together.
Function composition specifically involves chaining functions so that the output of one function becomes the input of another. This operation is intrinsic in many applications from physics to engineering because it allows the transformation of inputs through a series of processes.

• **Addition/Subtraction:** Combine functions by adding or subtracting their outputs.
• **Multiplication/Division:** Multiply or divide outputs of the functions.
• **Composition:** Use the output of a function as the input for the next.

Understanding these operations can promote skills in problem-solving and analysis across mathematics, as they allow for versatile manipulation and computation using functions in various configurations.