Problem 5 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of \(f \circ g(-2)\) is \(-12\).

Step 1: Understand Function Composition

The composition \(f \circ g\) means that we will first apply the function \(g\) to our input and then apply the function \(f\) to the result of \(g\). So, \((f \circ g)(x) = f(g(x))\).

Step 2: Apply the Inner Function \(g\)

Given \(g(x) = x - 2\), substitute \(-2\) for \(x\) to find \(g(-2)\).\[g(-2) = -2 - 2 = -4\]

Step 3: Apply the Outer Function \(f\)

Now use the result from Step 2 where \(g(-2) = -4\), and substitute \(-4\) into the function \(f(x) = 3x\).\[f(-4) = 3(-4) = -12\]

Step 4: Evaluate the Composition

Putting it all together, the result of the composition \(f \circ g(-2)\) is \[f(g(-2)) = f(-4) = -12.\]

Key Concepts

Understanding Functions

A function is a mathematical concept that defines a relationship between a set of inputs and outputs. In simple terms, a function takes an input, processes it, and gives an output. Think of it like a machine:

* You put in an ingredient (input),
* The machine processes it according to its rules (function),
* And then you get the final product (output).

In this exercise, we are given two functions, \(f(x) = 3x\) and \(g(x) = x - 2\).
* The function \(f(x)\), when given a number, multiplies it by 3.
* The function \(g(x)\) subtracts 2 from the input.

Functions represent how inputs change and can vary widely in complexity, from simple linear functions to complex algebraic ones.

The Art of Composition of Functions

The composition of functions involves combining two functions so that the output of one function becomes the input for the other. In mathematical notation, it's written as \(f \circ g\), read as "\(f\) composed with \(g\)." This is like having processes in series, where one process feeds directly into another.

Here's how it works:
* Start with an input value.
* Apply the first function (inside function \(g\)) and get an intermediate result.
* Plug this intermediate result into the second function (outside function \(f\)).

In our exercise, we applied \(g(x)\) first and found \(g(-2) = -4\). This result was then put into \(f(x)\) to find \(f(-4) = -12\). Thus, \((f \circ g)(-2) = -12\).

Function composition allows us to create more complex functions from simple ones, making it easier to model real-world processes that require multiple steps.

Exploring Algebraic Functions

Algebraic functions are a type of function defined using algebraic expressions. They include polynomials, radicals, and any combination of addition, subtraction, multiplication, and division of terms.

In our example, both \(f(x) = 3x\) and \(g(x) = x - 2\) are algebraic functions because they involve basic algebraic operations:
* \(f(x) = 3x\) is a linear function, which is the simplest type of polynomial where each term is of degree 1.
* \(g(x) = x - 2\) is also linear, essentially a shifted version of the identity function \(x\), subtracted by 2.

Algebraic functions allow us to predict and calculate values by plugging numbers into the formula, a core skill in both algebra and calculus. Studying these functions helps build a foundation for understanding more complex mathematical concepts.