Problem 25 - Algebra 2 and Trigonometry

Question

If \(\mathrm{h}(x)=2(x+1)\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x)),\) what are possible expressions for \(\mathrm{f}(x)\) and for \(\mathrm{g}(x) ?\)

Step-by-Step Solution

Answer

Functions can be \( f(x) = 2x \) and \( g(x) = x + 1 \).

Step 1: Identify Composition

The given functions suggest that \( h(x) = f(g(x)) \). We can consider one possible strategy for decomposing \( h(x) = 2(x+1) \) into two functions \( f \) and \( g \) where one is applied repeatedly on the other to reform \( h(x) \).

Step 2: Define Inner Function \( g(x) \)

There is flexibility in choosing \( g(x) \), but a logical choice is \( g(x) = x + 1 \). This simplification focuses on transforming \( x \) in the form of \( x+1 \).

Step 3: Define Outer Function \( f(x) \)

Once \( g(x) = x + 1 \) is determined, we need \( f(g(x)) \) to yield \( 2(x+1) \). If \( g(x) = x + 1 \), a straightforward choice for \( f(x) \) is \( f(x) = 2x \) because \( f(g(x)) = 2(g(x)) = 2(x+1) \).

Step 4: Verify the Composition

Verify that the choices \( g(x) = x + 1 \) and \( f(x) = 2x \) reconstruct \( h(x) \):* Substitute \( g(x) \) into \( f(x) \):\[ f(g(x)) = f(x+1) = 2(x+1) \]* See that this is precisely \( h(x) \). Therefore, \( f \) and \( g \) are correct.

Key Concepts

Function Decomposition

Function decomposition is a mathematical idea used to break down complex functions into simpler, individual functions. In the context of the provided exercise, we aim to separate the given function \( h(x) = 2(x+1) \) into two functions \( f(x) \) and \( g(x) \).
By doing so, we can express the original function in a different way that might be more insightful or easier to work with. This process is helpful in understanding how individual parts of a function can work together to form a complete algebraic expression.

In our solution, we start with the decision to set \( g(x) = x+1 \). This choice simplifies the number within the parentheses, isolating it as a separate function. The next step involves identifying \( f(x) \) in such a way that when it is applied to \( g(x) \), it mirrors the original function \( h(x) \).
The function \( f(x) = 2x \) serves this purpose, because we know that when \( f \) is applied to \( g \), i.e., \( f(g(x)) \), the outcome is indeed \( 2(x+1) \). This verification confirms that our decomposition of the function is accurate and effective.

Function Notation

Function notation is a conventional way of expressing functions in mathematics. Using function notation, we can express operations and the relationships between inputs and outputs clearly and efficiently.
In the exercise, function notation is used to express the composite function \( h(x) = f(g(x)) \). This implies that the function \( g(x) \) is first applied to the input \( x \), then \( f(x) \) acts on the result of \( g(x) \). This sequence is compactly represented in function notation as \( f(g(x)) \).

Using function notation has several advantages:

• Clarity of process: It specifies the order in which functions are applied, making it easier to follow the flow of operations.
• Comparison and analysis: By using different functions defined in notation, it's possible to compare their structures and impacts easily.

Understanding the role of each component in function notation helps to build a deeper insight into the overall function, allowing one to analyze and solve problems more efficiently.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition and multiplication). They are fundamental in representing mathematical ideas in a concise form.
In the exercise under discussion, the algebraic expression given is \( h(x) = 2(x+1) \). This expression tells us that, for any input \( x \), \( x+1 \) is first calculated, and then the result is multiplied by 2, as dictated by the expression.

To dissect this expression into simpler components, we view \( x+1 \) and \( 2 \), each as parts of the larger picture. Eventually, the decomposition results in two simpler functions: \( g(x) = x+1 \) and \( f(x) = 2x \).
Working with algebraic expressions like these helps us understand how smaller operations combine to produce a final outcome, lending valuable insight into the structure and transformation processes in algebra. Mastering algebraic expressions is key to solving a wide array of mathematical problems more efficiently.