Problem 101 - Precalculus : Building Concepts and Connections

Question

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Evaluate \((f \circ f)(2)\)

Step-by-Step Solution

Answer

The value of \((f \circ f)(2)\) is 22.

Step 1: Define Composite Function

First, define the composite function \(f \circ f\). According to the definition of a composite function, this means that we replace every instance of \(t\) in function \(f\) with the function \(f(t)\) itself. Therefore, \(f \circ f(t) = f(f(t)) = f(3t+1)\.

Step 2: Substitute and Simplify

Next, evaluate \(f \circ f(2)\) by substituting \(2\) for \(t\) in \(f(3t+1)\). This gives us \(f(3*2+1)\), which simplifies to \(f(7)\). The function \(f(t) = 3t+1\), so \(f(7) = 3*7+1 = 22\).

Step 3: Conclude

Therefore, \((f \circ f)(2) = 22\)

Key Concepts

Understanding Composite Functions

Composite functions involve combining two functions into one. To form a composite function, we take one function's output and use it as the input for another function. This results in what we call a composite function. For example, if we have two functions, say, \( f(t) = 3t + 1 \) and another, \( g(x) = x^2 + 4 \), the composition \( f \circ f \) is created by replacing the variable \( t \) in \( f(t) \) with the expression for \( f(t) \) itself:
- \( f \circ f(t) = f(f(t)) = f(3t + 1) \)
This manipulation essentially layers the functions, creating a more complex function from simpler ones.

Steps for Function Evaluation

Function evaluation involves substituting a given number into a function and calculating its output. It's essential to perform this step-by-step:

- **Step 1**: Begin with the defined composite function, such as \( f \circ f(t) = f(3t + 1) \).
- **Step 2**: Replace the variable \( t \) with the specific value you're evaluating, in this case, \( t = 2 \). This substitution ensures you're evaluating the composite function correctly. In this scenario, you substitute to get \( f(3 \times 2 + 1) = f(7) \).
- **Step 3**: Compute the value by using the inner function. For example, \( f(7) = 3 \times 7 + 1 = 22 \). This approach ensures you've accurately evaluated the function.

Mastering Algebraic Manipulation

Algebraic manipulation allows us to simplify and solve mathematical expressions efficiently. This skill involves rearranging terms and substituting values to simplify expressions or solve equations. When dealing with compositions, understanding the algebraic structure is crucial:

- **Substitute Clearly**: Carefully replace variables with their respective values or expressions to avoid errors. For instance, substitute \( 2 \) into \( f(3t+1) \) giving \( f(7) \).
- **Simplify Carefully**: Break down the expression into manageable steps. Compute \( f(7) \) by applying the function \( f(t) = 3t + 1 \) as understood. Every step logically follows the previous one and confirms accuracy by stabilizing unexpected variables.

Mastering these techniques makes tackling algebraic challenges in composite functions straightforward and less daunting.