Problem 27 - Algebra A Combined Function

Question

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ F(x)=9 x^{2}+2 $$

Step-by-Step Solution

Answer

The composition is \(F(x) = h(f(x))\).

Step 1: Understanding the Target Function

The function we need to express as a composition is \(F(x) = 9x^2 + 2\). Our task is to express this using the functions \(f(x), g(x),\) and \(h(x)\).

Step 2: Analyzing Component Functions

Identify the structure of the target function. Notice that \(F(x)\) resembles the form of \(h(x) = x^2 + 2\), except the \(x\) term is replaced with \(3x\). This suggests it might involve a composition of \(h(x)\) with another function.

Step 3: Express the Composition

To adjust \(h(x)\) such that it matches \(F(x) = 9x^2 + 2\), consider the expression \(9x^2\) which can be interpreted as \((3x)^2\). Thus, compose the function \(h(x)\) with \(f(x)\) such that \(h(f(x))\) is evaluated.

Step 4: Verifying the Composition

Calculate \(h(f(x))\) to verify:\[ h(f(x)) = h(3x) = (3x)^2 + 2 \]Simplifying the right side gives \(9x^2 + 2\), which matches \(F(x)\). Therefore, \(F(x)\) can be expressed as \(h(f(x))\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions constructed using a combination of basic operations such as addition, subtraction, multiplication, division, and root extraction. These functions play a vital role in algebra by helping us map inputs to outputs using specific rules. In this context:

• The function \( f(x) = 3x \) is a linear function because it is represented by a straight line when graphed. Its slope is 3, meaning for each unit increase in \( x \), \( f(x) \) increases by 3 times that value.
• The function \( g(x) = \sqrt{x} \) is a radical function, which involves the square root of \( x \). It is defined only for non-negative values of \( x \), as square roots of negative numbers are not real.
• The function \( h(x) = x^2 + 2 \) is a quadratic function, signifying its term with \( x^2 \). Such functions form a parabolic shape when plotted, characterized by a curve opening upwards.

By using these algebraic functions, one can represent and solve many practical problems in mathematics.

Function Analysis

Function analysis involves dissecting a function to understand its properties, behavior, and structure. By examining the function's form, analysts gain insights into how the function behaves across different input values. For example, in our case:

• The target function \( F(x) = 9x^2 + 2 \) suggests elements of both operations on \( x \) due to its quadratic term \( 9x^2 \) and the constant \( +2 \).
• Comparing \( F(x) \) with \( h(x) = x^2 + 2 \), we see that the function inherits a similar structure, except the \( x \) in \( x^2 \) is replaced by \( 3x \), indicating a scaling factor of 3.
• Such analysis leads us to hypothesize that \( F(x) \) is formed by a composition of functions, where the input is modified by another function before being fed into \( h(x) \).

Analyzing functions in this way is crucial for correctly expressing them in terms of simpler component functions, making them easier to understand and evaluate.

Composite Functions

Composite functions are functions created by combining two or more functions, where the output of one function becomes the input of the next. This is often denoted as \( (f \circ g)(x) = f(g(x)) \), which means \( g(x) \) is performed first and its result becomes the input for \( f(x) \). In the composition for our problem:

• We analyze \( F(x) = 9x^2 + 2 \) and recognize it can be expressed as the composite function \( h(f(x)) \), where \( f(x) = 3x \) is first applied to \( x \). The result \( 3x \) is then used as input for \( h(x) \).
• This illustrates the concept of nested functions and how altering the input function can directly affect the overall answer. For instance, applying \( h \) to \( f(x) = 3x \) leads to \((3x)^2 + 2\), which matches the final expression of \( F(x) \).
• Working with composite functions helps in breaking down complex functions into smaller, more manageable computations, easing both their evaluation and understanding.

Understanding composite functions is key to solving many issues in mathematics, particularly when tackling complex function problems like the one presented here.