Problem 42 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Composition

The symbol \((g \circ f)(x)\) represents the composition of two functions, where we first apply \(f(x)\) and then apply \(g(x)\) to the result. Therefore, \((g \circ f)(x) = g(f(x))\). Our task is to find \((g \circ f)\left(\frac{1}{3}\right)\).

Step 2: Apply the Function \(f(x)\)

Start by applying \(f\) to \(\frac{1}{3}\). Substitute \(\frac{1}{3}\) into \(f(x) = 2x + 1\):\[ f\left(\frac{1}{3}\right) = 2\left(\frac{1}{3}\right) + 1 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]

Step 3: Apply the Function \(g(x)\)

Now substitute the result from Step 2 into \(g(x) = x^2 - 1\): \[ g\left(\frac{5}{3}\right) = \left(\frac{5}{3}\right)^2 - 1 \]Calculate \(\left(\frac{5}{3}\right)^2 = \frac{25}{9}\) and then substitute back to find:\[ g\left(\frac{5}{3}\right) = \frac{25}{9} - 1 = \frac{25}{9} - \frac{9}{9} = \frac{16}{9} \]

Step 4: Compose the Functions

Finally, compose the results from Step 2 and Step 3. We found that the composition \((g \circ f)\left(\frac{1}{3}\right)\) gives us the value of \(g(f(x))\) for \(x = \frac{1}{3}\):\[ (g \circ f)\left(\frac{1}{3}\right) = g\left(f\left(\frac{1}{3}\right)\right) = g\left(\frac{5}{3}\right) = \frac{16}{9} \]

Key Concepts

Algebra

Algebra is all about using letters and symbols to represent numbers and quantities in formulas and equations. In this exercise, we are working with functions that are part of algebra, specifically the manipulation and operation of these functions to solve problems. It's important to keep in mind the basic rules of algebra as they allow us to solve equations step-by-step effectively. You might encounter different operations such as addition, subtraction, multiplication, and division along with powers and square roots. As you practice more problems, these rules will become second nature.

Function Evaluation

Evaluating a function means calculating the output value of a function for a given input. In our exercise, we begin with the function \( f(x) = 2x + 1 \). The task is to evaluate this function for the input \( x = \frac{1}{3} \). Here are the steps:

• Substitute \( \frac{1}{3} \) directly into \( f(x) \).
• Perform the arithmetic: \( f\left(\frac{1}{3}\right) = 2\times\frac{1}{3} + 1 \).
• Simplify this expression, \( 2\times\frac{1}{3} = \frac{2}{3} \), so the result becomes \( \frac{2}{3} + 1 \).
• Convert \( 1 \) to \( \frac{3}{3} \) to get common denominators, resulting in \( \frac{2}{3} + \frac{3}{3} = \frac{5}{3}\).

Function evaluation is crucial as it helps us understand the relationship between inputs and outputs within a function.

Step-by-Step Solution

This exercise is solved through a sequence of clear and logical steps. Understanding each of these steps individually helps in grasping the bigger picture. Here’s a breakdown:Firstly, understand the notion of **function composition**, which is applying one function to the results of another, i.e., we apply \( f(x) \) and then apply \( g(x) \) to the result, written as \( g(f(x)) \). For the expression \( (g \circ f)(\frac{1}{3}) \), it means substituting \( x \) into \( f \), and then using that result in \( g \).Steps to Solve:

• Evaluate \( f\left(\frac{1}{3}\right) \) as previously detailed, resulting in \( \frac{5}{3} \).
• Next, substitute this \( \frac{5}{3} \) into the function \( g(x) = x^2 - 1 \).
• Calculate \( g\left(\frac{5}{3}\right) = \left(\frac{5}{3}\right)^2 - 1 \).
• Work out \( \left(\frac{5}{3}\right)^2 = \frac{25}{9} \) and simplify to get \( \frac{16}{9} \).

This careful sequencing of steps ensures you are always applying functions in the right order, leading to the correct solution: \( \frac{16}{9} \). As you practice, going step-by-step prevents mistakes and helps reinforce your understanding of function compositions.