Problem 22 - Algebra and Trigonometry

Question

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{lll}{\text { (a) } f(f(4))} & {\text { (b) } g(g(3))}\end{array} $$

Step-by-Step Solution

Answer

(a) 16, (b) -47

Step 1: Evaluate Inner Function for Part (a)

First, we need to evaluate the inner function for part (a), which is \(f(4)\). Plug \(x = 4\) into the function \(f(x) = 3x - 5\). Calculate: \(f(4) = 3(4) - 5 = 12 - 5 = 7\).

Step 2: Evaluate Outer Function for Part (a)

With the result from Step 1, evaluate the outer function \(f(f(4))\), which is \(f(7)\). Use the function \(f(x) = 3x - 5\), plug in \(x = 7\), and calculate: \(f(7) = 3(7) - 5 = 21 - 5 = 16\).

Step 3: Evaluate Inner Function for Part (b)

Next, evaluate the inner function for part (b), which is \(g(3)\). Plug \(x = 3\) into the function \(g(x) = 2 - x^2\). Calculate: \(g(3) = 2 - (3)^2 = 2 - 9 = -7\).

Step 4: Evaluate Outer Function for Part (b)

Now, evaluate the outer function \(g(g(3))\) using the result from Step 3: \(g(-7)\). Use the function \(g(x) = 2 - x^2\), plug in \(x = -7\), and calculate: \(g(-7) = 2 - (-7)^2 = 2 - 49 = -47\).

Key Concepts

Inner Function

When solving problems involving function composition, understanding the concept of the 'inner function' is crucial. Imagine your journey through a math problem as navigating through layers. The inner function is like the first layer you encounter and interact with.

• **Definition:** It's the function you evaluate first from the inside out.
• **Example in Practice:** In the problem given, for part (a), the inner function is evaluated when plugging 4 into the function \( f(x) = 3x - 5 \). This is expressed as \( f(4) \).

The goal here is to simplify the expression by calculating the inner function's value. Once this value is found, it becomes the input for the next layer, which is the outer function. It’s essential that this step is completed accurately, as any mistake will affect the final outcome. Remember, in mathematical terms, the inner function addresses the direct input value or variable, initiating the process of finding the final solution.

Outer Function

When we move from the inner function to the outer function, it's like stepping into the next phase of a journey. The outer function takes the result from the inner function and continues the computation process.

• **Definition:** The outer function is the function applied second or last, using the result from the inner function as its input.
• **Example in Practice:** In part (a) of the exercise, after determining that \( f(4) = 7 \), the outer function is \( f(f(4)) = f(7) \). This is calculated using the same function as before: \( f(x) = 3x - 5 \).

This stage is critical as it produces the final answer to the composition problem. The outer function is essentially the 'goal' of the function evaluation, where all interpretations and simplifications lead. Keep in mind that accuracy in evaluating the inner function will have a direct impact on this step, making its correct execution vital for a successful solution.

Function Evaluation

Function evaluation is the process of computing the output of a function given an input value. This is an essential skill in mathematics, especially when dealing with compositions of functions.

• **Understanding the Process:** Both inner and outer functions are evaluated through plugging in concrete values for variables and performing arithmetic operations as defined by the function's rule.
• **Example in the Exercise:** For \( f(x) = 3x - 5 \), when you input \( x = 4 \), you do the arithmetic \( 3 \times 4 - 5 \) to find the result of \( f(4) \). For \( g(x) = 2 - x^2 \), performing \( g(3) \) involves calculating \( 2 - 3^2 \), which results in \(-7\).

Function evaluation ensures that each operation follows the rules provided by the function's definition. It requires attention to detail and precise calculations. Missteps can lead to incorrect outcomes, so each evaluation must be done carefully, ensuring that each subsequent computation is based on correct preceding results. Through practice, these steps can become more intuitive, aiding in solving more complex equations with confidence.