Problem 28 - 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 \(h .\) $$ H(x)=3 x^{2}+6 $$

Step-by-Step Solution

Answer

H(x) can be expressed as 3 times the composition of h(x).

Step 1: Analyze the given function

The function we have is \( H(x) = 3x^2 + 6 \). We want to express this function as a composition of \( f(x) = 3x \), \( g(x) = \sqrt{x} \), or \( h(x) = x^2 + 2 \).

Step 2: Compare with known functions

Notice the form of \( H(x) = 3x^2 + 6 \). This expression resembles the function \( h(x) = x^2 + 2 \) but with different coefficients.

Step 3: Find a way to express H(x) using compositions

Notice that \( 3x^2 = 3 \cdot (x^2) \), which suggests using \( h(x) \) since it involves \( x^2 \). So, if we substitute \( x^2 \) from \( h(x) \) and multiply by 3, we arrive at \( 3h(x) = 3(x^2 + 2) = 3x^2 + 6 \).

Step 4: Combine to express as a composition

The function \( H(x) = 3x^2 + 6 \) can now be considered as using \( 3 \times h(x) \). The replacement within this composite affects \( h(x) = x^2+2 \) which completes the given requirements.

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions constructed using algebraic operations such as addition, subtraction, multiplication, division, and root extraction. These functions are incredibly flexible and can describe a wide range of relationships and transformations. For example, in our case, we have functions like \(f(x) = 3x\), an example of a simple linear algebraic function, \(g(x) = \sqrt{x}\), a radical function, and \(h(x) = x^2 + 2\), a quadratic function. Each of these functions can be combined with operations like composing with each other to formulate new expressions. Algebraic functions serve as the building blocks for more complex mathematical operations and can elucidate the relationship between different algebra expressions.

Composition of Functions

The composition of functions involves taking two or more functions and combining them in a way that the output of one function becomes the input of another. If you have a function \(f(x)\) and another function \(h(x)\), the composition \((f \circ h)(x)\) effectively means that you are taking the output from \(h(x)\) as the input to \(f(x)\). In the exercise provided, we are essentially exploring how \(H(x) = 3x^2 + 6\) can be rewritten using the functions \(f(x)\), \(g(x)\), or \(h(x)\). We observed that \(h(x) = x^2 + 2\) is very close in structure to \(H(x)\) and by appropriately scaling it by a factor of 3, we align our function \(H(x)\) with \(3h(x)\), which illustrates the concept of composition quite distinctly.

Mathematical Expressions

Mathematical expressions represent formulas that combine numbers, variables, and operations. In the given problem, the expression \(H(x) = 3x^2 + 6\) is a polynomial expression, commonly used in algebra to characterize functions graphically and numerically. The transformation of such expressions into different forms involves recognizing patterns and substituting parts of one expression with equivalent elements from another. In mathematical expressions, constants, and coefficients like "3" in our example remain crucial in dictating the changes in form and scale. Understanding how substitutions and compositions work within equations helps clarify how to form equivalent expressions, which can sometimes reveal insights into the underlying problem or simplify complex calculations.