Problem 44 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((f \circ g)(2x) = 8x^2 - 1\).

Step 1: Understand Function Composition

In this exercise, the notation \((f \circ g)(x)\) denotes the composition of the functions \(f\) and \(g\), which means that \((f \circ g)(x) = f(g(x))\). The task is to find \((f \circ g)(2x)\). This involves evaluating \(g(2x)\) first and then using the result to find \(f\).

Step 2: Calculate \(g(2x)\)

To find \(g(2x)\), substitute \(2x\) into the function \(g(x) = x^2 - 1\). This yields:\[g(2x) = (2x)^2 - 1 = 4x^2 - 1.\]

Step 3: Calculate \(f(g(2x))\)

Now that we have \(g(2x) = 4x^2 - 1\), substitute this result into \(f(x) = 2x + 1\). Thus, substitute \(4x^2 - 1\) for \(x\) in \(f(x)\):\[f(4x^2 - 1) = 2(4x^2 - 1) + 1.\] Simplify to get:\[f(4x^2 - 1) = 8x^2 - 2 + 1 = 8x^2 - 1.\]

Step 4: Final Result

The composition of the functions \((f \circ g)(2x)\) has been found to be:\[(f \circ g)(2x) = 8x^2 - 1.\]

Key Concepts

Composition of Functions

Function composition is like building a machine where the output of one function becomes the input of another. In mathematical notation, when you see \((f \circ g)(x)\), it represents the composition of two functions \(f\) and \(g\). This means you first evaluate \(g(x)\) and then use its result as the input for \(f(x)\). Consider it as a two-step process:

• Start with \(g\) and calculate \(g(x)\).
• Take the output from \(g(x)\) and plug it into \(f\), resulting in \(f(g(x))\).

For example, in our exercise, \((f \circ g)(2x) = f(g(2x))\). You first find \(g(2x)\) and then use that to find \(f\), resulting in a complete composition. This method is crucial for solving problems involving function transformations.

Evaluate Functions

Evaluating functions is about substituting a given input into an equation to find the output. It’s similar to placing an ingredient in a recipe, where the function decides how the input is processed to get a result.

For the function \(g(x) = x^2 - 1\), to evaluate \(g(2x)\), you simply replace every \(x\) with \(2x\). This gives \(g(2x) = (2x)^2 - 1\). Calculating further, this results in \(4x^2 - 1\). Similarly, for the function \(f(x) = 2x + 1\), if you substitute \(4x^2 - 1\) (which is \(g(2x)\)), you evaluate \(f(4x^2 - 1)\). The substitution process is:

• Replace \(x\) in \(f(x)\) with \(4x^2 - 1\).
• Calculate \(2(4x^2 - 1) + 1\).
• Simplify to obtain \(8x^2 - 1\).

Evaluating functions is a straightforward way to find outputs for specific inputs, helping you understand how functions interact with values.

Algebraic Expression Simplification

Simplification involves reducing expressions to their simplest form, making them easier to understand and work with. It’s much like cleaning up a cluttered room, where you eliminate unnecessary items.

Take the expression \(f(4x^2 - 1) = 2(4x^2 - 1) + 1\). To simplify this:

• First, distribute \(2\) across terms inside the parentheses: \(2 \times 4x^2 - 2 \times 1\), giving \(8x^2 - 2\).
• Then, add \(1\) to \(-2\), resulting in \(8x^2 - 1\).

Simplification removes complexity, allowing us to better visualize and solve mathematical problems. This process is vital when working through function composition, as it clarifies the final solution and ensures accuracy.