Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{f}(\mathrm{g}(4)) $$

Step-by-Step Solution

Answer

The value of \( f(g(4)) \) is 6.

Step 1: Evaluate Inner Function

First, we need to find the value of the inner function, which is \( g(x) \). Substitute \( x = 4 \) into \( g(x) = x - 2 \). Thus, \( g(4) = 4 - 2 = 2 \).

Step 2: Evaluate Outer Function

Now, take the result from Step 1 and use it as input for the outer function \( f(x) \). Substitute \( g(4) = 2 \) into \( f(x) = 3x \). Thus, \( f(g(4)) = f(2) = 3 \times 2 = 6 \).

Step 3: Confirm the Composition Result

Verify that we followed the correct order: evaluated \( g(x) \) first and then \( f(x) \) using the result from \( g(x) \). The final result for \( f(g(4)) \) is \( 6 \).

Key Concepts

Functions Evaluation

Function evaluation may seem like an intimidating concept, but it's quite simple when you break it down. Imagine a function as a machine: you put something in, and it gives you something out. For functions, we input a specific number, and a unique output comes out based on the rules of the function.

Let’s consider the function, say, \( f(x) = 3x \). Here, \( x \) is like the slot where you can input any number. To evaluate \( f(x) \), replace \( x \) with the specific number you're given. For instance, if \( x = 2 \), then \( f(2) = 3 \cdot 2 = 6 \). You can use this same idea for any number you need to evaluate the function with.

For function composition, you'll be dealing with two or more functions. One function's output becomes another function's input. So, you'll evaluate the inner function first, just like in a nested puzzle. It’s kind of like baking a cake and then putting icing on top – you have to bake the cake first before you can ice it!

Algebraic Functions

Algebraic functions are fundamental in mathematics. These are functions built using operations like addition, subtraction, multiplication, division, and exponentiation, among others. Think of them as recipes where specific ingredients (numbers and operations) come together to yield a desired result.

For example, both \( f(x) = 3x \) and \( g(x) = x-2 \) in our original exercise are algebraic functions. They use simple operations:

• \( f(x) = 3x \) involves multiplication.
• \( g(x) = x - 2 \) involves subtraction.

These operations combine inputs to give an output. Algebraic functions can be quite simple, like the ones in our exercise, or very complex based on how many operations and variables are involved. Regardless of complexity, they follow consistent mathematical rules that make calculations predictable and reliable.

Step by Step Solutions

Working through a problem step by step is a powerful learning tool. It not only aids in understanding but also helps in pinpointing where you might go wrong. For function composition, specifically, following the proper order is crucial. You initially find out what the inner function evaluates to, and then turn to the outer function.

In our example, we have the composition \( f(g(x)) \). This means we first tackle \( g \), by substituting \( x \) with 4 in \( g(x) = x - 2 \), resulting in \( g(4) = 2 \). Then, that result becomes the input in the next step for \( f(x) = 3x \). The process continues by substituting to get \( f(2) = 3 \times 2 = 6 \).

Upon completion, always compare every single step to what's expected. By confirming each step, you ensure the solution is accurate. The method of meticulously following through each part of the problem is vital for successful evaluations and showcases the beauty of mathematical logical order.