Problem 86 - Algebra 2

Question

If \(f(x)=2 x^{2}\) and \(g(x)=3 x,\) what is \((g \circ f)(x) ?\) $$\begin{array}{llll}{\text { F. } 6 x^{2}} & {\text { G. } 9 x^{2}} & {\text { H. } 18 x^{2}} & {\text { J. } 8 x^{4}}\end{array}$$

Step-by-Step Solution

Answer

The composite function \((g \circ f)(x)= 6x^{2}\).

Step 1: Understand What is Given

We are given two functions \(f(x)=2 x^{2}\) and \(g(x)=3 x\), and asked to find the composite function \((g \circ f)(x)\).

Step 2: Evaluate the Composition of the Functions

The notation \(g \circ f(x)\) means that we replace the input of function \(g\) with the output from function \(f\). Thus, we substitute \(f(x)=2 x^{2}\) into \(g(x)\), replacing every x in g with the formula for \(f(x)\). Doing this gives us \(g(f(x)) = g(2x^{2})\). Now, we substitute 2x^2 into g(x) to get the final composite function.

Step 3: Finalize and Simplify the Composite Function

Substitute \(2x^{2}\) into \(g(x)=3x\), we will get \(3(2x^{2}) = 6x^{2}\). So, \((g \circ f)(x) = 6x^{2}\).

Key Concepts

Understanding Function Notation

Function notation is a crucial concept in algebra that helps in denoting and working with functions in an organized manner. It's a way of expressing functions using symbols and letters. When you see something like \(f(x)\), it refers to a function named \(f\) with \(x\) as its variable.

• \(f(x)\) indicates the output of the function when \(x\) is the input.
• The letter \(f\) could be any letter, like \(g\), \(h\), or even Greek letters like \(\phi\).

Function notation allows us to work with complex relationships step by step, making calculations manageable. In the example we have, \(f(x) = 2x^2\) tells us that if you plug a number into \(x\), you square it and then multiply by 2 to find the output. This is a neat and concise way to communicate mathematical relationships.

Basics of Quadratic Functions

A quadratic function is a type of polynomial function where the highest degree of any variable is 2. It is commonly expressed as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Here are a few points to remember:

• The graph of a quadratic function is a parabola, which can open upwards or downwards.
• The vertex of the parabola can give valuable information about the maximum or minimum point.

In the given problem, \(f(x) = 2x^2\) is a quadratic function where the coefficient \(a = 2\). This tells us that every input \(x\), once squared, is multiplied by 2 to get the output. Understanding the behavior of a parabola helps in visualizing the function's path and nature.

Working with Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. They are the building blocks of equations and functions. In this context, we deal with expressions like \(3x\) or \(2x^2\). Here is what you need to know:

• Expressions can be simplified by performing operations such as addition, subtraction, multiplication, and division.
• Substitution plays a vital role in evaluating expressions, especially in function composition.

In our exercise, composing functions involves substituting one expression into another. For instance, inserting \(f(x) = 2x^2\) into \(g(x) = 3x\), makes us replace \(x\) in \(g(x)\) with \(2x^2\). This results in the new expression \(3(2x^2)\), simplifying to \(6x^2\). Mastery of algebraic expressions is key to solving such problems efficiently.