Problem 65 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=3 x^{2}+2, g(x)=2 x $$

Step-by-Step Solution

Answer

\(f(g(x)) = 12x^2 + 2, g(f(x)) = 6x^2 + 4\).

Step 1: Compute \(f(g(x))\)

We begin by substituting \(g(x) = 2x\) into the function \(f(x)\) where \(x\) appears. Thus, we get: \(f(g(x)) = f(2x) = 3(2x)^2 + 2\).

Step 2: Simplify \(f(g(x))\)

Now we perform the operations in the function and simplify: \(f(g(x)) = 3*4x^2 +2 = 12x^2 + 2\).

Step 3: Compute \(g(f(x))\)

Next, we substitute \(f(x) = 3x^2 + 2\) into the function \(g(x)\) where \(x\) appears. So, we obtain: \(g(f(x)) = g(3x^2 + 2) = 2(3x^2 + 2)\).

Step 4: Simplify \(g(f(x))\)

Finally, we carry out the operations in the function and simplify: \(g(f(x)) = 6x^2 + 4\).

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental in math, combining numbers, variables, and mathematical operations. An expression could be as simple as a number, like 5, or it may include variables like \(x\) and operations like multiplication or addition.

In the context of function composition, understanding how to interpret and manipulate these expressions is essential. For instance, for the function \(f(x) = 3x^2 + 2\), this expression includes the term \(3x^2\), which signifies that \(x\) is squared and then multiplied by 3. The constant 2 is added at the end. Each part of the expression plays a role and determines how the function behaves.

When dealing with functions, here are important points to remember about algebraic expressions:

• Each term in an expression is a separate piece that adds up to the whole.
• The coefficient is the number multiplying the variable term, like the 3 in \(3x^2\).
• Exponents, like the 2 in \(x^2\), indicate repeated multiplication.

Substitution Method

The substitution method is used to replace variables in an expression or function with specific values. This is crucial when composing functions. For instance, when given functions \(f(x)\) and \(g(x)\), substitution allows us to evaluate expressions like \(f(g(x))\).

In function composition, the process happens as follows:

• First, we identify the inner function, in this example, it is \(g(x)\).
• Then, wherever the variable appears in the outer function \(f(x)\), we replace it with the entire expression of \(g(x)\). Here, it involves replacing \(x\) in \(f(x) = 3x^2 + 2\) with \(2x\) from \(g(x)\).

This substitution changes the variables of the original functions, combining them into a new expression. Such transformations are routine in algebra and integral in many areas, including calculus and differential equations.

Function Operations

Function operations involve performing mathematical processes on functions. Operations such as addition, subtraction, multiplication, division, and composition can expand our toolkit for handling complex expressions.

Let's discuss composition, which is a key operation when dealing with functions like \(f(x)\) and \(g(x)\). Function composition implies feeding one function into another, such as \(f(g(x))\) or \(g(f(x))\).

Here’s how it works:

• Start by identifying the functions you'll compose.
• Replace the variable in the "outer" function with the expression of the "inner" function.
• Simplify the resulting expression by performing the indicated operations. This helps transform and reduce the expression into simpler forms.

For instance, when composing \(f(g(x)) = 3(2x)^2 + 2\), performing operations allows us to simplify this to \(12x^2 + 2\). Function operations help us create powerful expressions suitable for modeling and solving real-world problems.