Problem 71 - 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.) $$h(x)=(2 x+1)^{2}$$

Step-by-Step Solution

Answer

The functions that satisfy \((f \circ g)(x) = h(x)\) are \(f(x) = x^{2}\) and \(g(x) = 2x + 1\).

Step 1: Identify function \(g(x)\)

We need the inner function first which is called \(g(x)\). This function should result in the expression inside the square brackets in \(h(x)\). Let's take \(g(x) = 2x+1\), since this is the function within \(h(x)\).

Step 2: Identify function \(f(x)\)

Now, we look at the overall function of \(h(x)\). We know this is the function \(f(x)\) composed with \(g(x)\), meaning \(f(g(x))\). So, the function \(f(x)\) must be \(f(x) = x^{2}\).

Step 3: Verify

Let's verify. The composition of \(f\) and \(g\) becomes \(f(g(x)) = f(2x+1) = (2x+1)^{2}\) which matches exactly with \(h(x)\).

Key Concepts

Composite Functions

When studying functions in mathematics, one fascinating concept is the creation of composite functions. A composite function is the result of combining two functions, denoted as \(f \circ g)(x)\), where the output of one function becomes the input of the other. To picture this, imagine two machines: the first machine takes a raw material and transforms it into a semi-finished product (\(g(x)\)), and then the second machine takes this semi-finished product and completes it to a finished product (\(f(x)\)).

In mathematical terms, the function \(g\) is applied to \(x\) first, and then function \(f\) is applied to the result of \(g(x)\). This process can be seen as a function transformation journey, where \(x\) undergoes multiple modifications. For the expression \(h(x) = (2x+1)^2\), a possible pair of functions that combine to form \(h\) could be \(g(x) = 2x+1\) and \(f(x) = x^2\). Such flexibility allows various functions to be creatively paired to achieve the desired composite function.

Inner and Outer Functions

Delving deeper into the composition of functions, we meet the concepts of inner and outer functions. Much like a nested doll, where one doll is encased within another, the inner function \(g\) is encapsulated within the outer function \(f\).

The inner function \(g(x)\) is applied first; it's the 'raw material' we mentioned. It holds the initial pattern or rule that shapes our input. The outer function \(f(x)\), on the other hand, is like the final craftsman, applying its rule to the result of \(g(x)\). Returning to our problem, \(g(x) = 2x+1\) is the inner function, as it provides the first transformation for \(x\). Subsequently, \(f(x) = x^2\) is the outer function that squares the output of \(g(x)\). Understanding this relationship helps us visualize and perform compositions more effectively, creating a bridge between the two functions for desired outcomes.

Verifying Function Composition

After constructing composite functions, it's crucial to verify that the composition accurately reflects the target function. This step ensures our inner and outer functions interact correctly to produce the complex behavior of the composite function.

To verify a function composition, first, apply the inner function \(g(x)\), then take its result and apply the outer function \(f\) to it. This process is symbolically represented as \(f(g(x))\). If the final output matches the given composite function \(h(x)\), our choice was correct. From the exercise, we verify that \(f \circ g)(x) = f(g(x)) = f(2x+1) = (2x+1)^2\), which perfectly aligns with our target function \(h(x)\). Verification reassures that our functions \(f\) and \(g\) have been accurately paired. This method of verification is a powerful tool for ensuring accuracy in mathematics and other areas requiring precise function manipulation.