Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=(x+4)^{2}+2(x+4)$$

Step-by-Step Solution

Answer

The functions \(f(x) = x^{2}+2x\) and \(g(x) = x+4\) make up the given function \(h(x)\) through composition, i.e., \((f \circ g)(x)=h(x)\)

Step 1: Identify g(x)

Identify a function within \(h(x)\) that can play the role of \(g(x)\). In this case, we can take \(g(x) = x+4\) as it appears twice in the equation \(h(x)=(x+4)^{2}+2(x+4)\).

Step 2: Identify f(x)

Identify \(f(x)\) so that it modifies \(g(x)\) to produce \(h(x)\). We have the square of \(g(x)\) plus double of \(g(x)\) in \(h(x)\). Therefore, we can take \(f(x) = x^2 + 2x\).

Step 3: Verify the solution

Finally, we need to verify that \((f \circ g)(x) = h(x)\). This means that, substituting \(g(x) = x+4\) into \(f(x)\) should produce \(h(x)\). Therefore, \(f(g(x)) = (x+4)^{2}+2(x+4)\), which is indeed the given function \(h(x)\).

Key Concepts

Function Decomposition

Function decomposition is like unraveling a complex math function into its simpler parts. It involves breaking down a complicated function into two or more simpler functions that together create the original complex function. This is very helpful for understanding and solving problems involving functions, especially when dealing with complex equations.
In our example, we have the function \(h(x) = (x+4)^2 + 2(x+4)\), which can be decomposed into two simpler functions: \(f(x)\) and \(g(x)\). We chose \(g(x) = x + 4\) because \(x + 4\) appears repeatedly in the equation.
Then, we chose \(f(x) = x^2 + 2x\), which perfectly utilizes the output of \(g(x)\) when substituted back into \(h(x)\). Decomposing functions in this way allows us to simplify the process of evaluating or manipulating them, making the function easier to understand and handle.

Composite Functions

Composite functions are like layers, where the output of one function becomes the input of another. Imagine them as machines in a sequence: one machine processes the input, then passes the result to the next machine. This is what happens with function composition.
In mathematical terms, when you have two functions \(f(x)\) and \(g(x)\), and you want to create a composite function \((f \circ g)(x)\), it means you first apply \(g\) to \(x\), and then apply \(f\) to the result of \(g(x)\). This is why it's called a composite—it combines both actions into one.
For this exercise, we demonstrated this by applying \(g(x) = x + 4\) first and then applying \(f\) to get \(f(g(x)) = (x+4)^2 + 2(x+4)\). Composite functions offer a powerful way of building more complex functions from simpler ones.

Elementary Functions

Elementary functions are the basic building blocks of all other functions. They are simple and common functions like polynomials, exponentials, logarithms, and trigonometric functions. Think of them as the alphabet of functions.
In our case, we used an elementary polynomial function \(x^2 + 2x\) for \(f(x)\) and another simple linear function \(x + 4\) for \(g(x)\). These are both elementary because they involve basic operations like addition, multiplication, and raising a term to a power.
Because these functions are straightforward, they are easy to work with and combine to form more complex functions. Recognizing elementary functions within a problem makes it a lot easier to decompose or compose functions, as we did in this exercise.