Problem 30 - Algebra Form and Function

Question

If \(f(g(x))=5\left(x^{2}+1\right)^{3}\) and \(g(x)=x^{2}+1\), find \(f(x)\).

Step-by-Step Solution

Answer

Answer: The function \(f(x)\) is \(f(x) = 5\left((x^2 + 1)^2 + 1\right)^3\).

Step 1: Rewrite the composite function with the given function \(g(x)\)

First, let's rewrite the composite function \(f(g(x))\), with the given function \(g(x)\), which is \(x^2 + 1\). So we have \(f(g(x)) = 5\left((x^2 + 1)^2 + 1\right)^3\).

Step 2: Substitute \(g(x)\) with a placeholder variable, \(y\).

Now we will replace every instance of \(g(x)\) with a new variable \(y\). We will then express \(f(g(x))\) entirely in terms of this new variable \(y\). So, replacing \(x^2 + 1\) with \(y\), we get \(f(y) = 5(y^2 + 1)^3\).

Step 3: Solve for \(f(x)\)

The final step is to substitute back the expression of \(g(x)\) we used as a placeholder (i.e., \(y = x^2 + 1\)) into our simplified equation for \(f(y)\). So, replacing \(y\) with \(x^2 + 1\), we obtain: \(f(x) = 5\left((x^2 + 1)^2 + 1\right)^3\). Therefore, we have found the function \(f(x) = 5\left((x^2 + 1)^2 + 1\right)^3\).

Key Concepts

Function Notation

Function notation is a way to succinctly denote and work with functions. In mathematics, we use the letter, typically "f", to denote a function, and the letter is often followed by a variable in parentheses, like "f(x)", to show the input variable. This notation helps in understanding how functions depend on their inputs. For example, if we have a function notation such as \(f(x) = x^2\), it tells us that the function named \(f\) transforms input \(x\) into \(x^2\).

Using function notation is also crucial when dealing with composite functions, which are functions formed by combining two or more functions. The exercise involves a composite function \(f(g(x))\), implying that \(g(x)\) is first applied to \(x\), and then \(f\) is applied to the result of \(g(x)\). Understanding function notation is key to solving problems like this. It's all about tracking how inputs are transformed within a function and how functions can be nested within each other.

Function Substitution

Function substitution is a vital technique when dealing with composite functions. It involves substituting one function into another, replacing the inner function with a placeholder or a specific variable.

Let's look at our example: \(f(g(x)) = 5((x^2 + 1)^2 + 1)^3\). Since \(g(x) = x^2 +1\), one way to work through this is by substituting \(g(x)\) with a placeholder variable, let's call it \(y\). That means everywhere we see \(g(x)\), we instead write \(y\).

So, if \(f(g(x))\) becomes \(f(y) = 5(y^2 + 1)^3\), we've temporarily simplified the expression and can work on solving for \(f(y)\). We eventually substitute back \(y = x^2 + 1\) to determine \(f(x)\). This step is crucial for simplifying complex expressions and making the problem more manageable.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to find a desired outcome. It's all about using rules of algebra to transform equations into more workable forms. This skill is essential in mathematics, particularly when you need to isolate a particular function or variable.

In our example, once we replace \(g(x)\) with a placeholder \(y\), we perform algebraic manipulation on \(f(y) = 5(y^2 + 1)^3\). This expression shows us how \(f(y)\) depends on \(y\).

To solve for \(f(x)\), we'd substitute back \(y\) with \(x^2 + 1\). So we transform the function to its form \(f(x) = 5((x^2 + 1)^2 + 1)^3\).

Algebraic manipulation helps in breaking down these relationships and finding the final simplified form of the function needed to solve the problem. It transforms complex algebraic structures into simpler, solvable pieces.