Problem 41 - Contemporary Precalculus

Question

Write the given function as the composite of two functions, neither of which is the identity function, as in Examples 6 and 7 . (There may be more than one way to do this.) $$h(x)=\left(7 x^{3}-10 x+17\right)^{7}$$

Step-by-Step Solution

Answer

In the function $$h(x)=\left(7 x^{3}-10 x+17\right)^{7}$$, the two functions that create this composite function are $$f(x) = 7x^3 - 10x + 17$$ and $$g(x) = x^7$$.

Step 1: Identify the inner function

First, let's identify the inner function that is being raised to the power of 7. In this case, it is the cubic polynomial $$7x^3 - 10x + 17$$. Therefore, we set $$f(x) = 7x^3 - 10x + 17$$ as our first function.

Step 2: Identify the outer function

The outer function is responsible for raising the inner function to the power of 7. In mathematical terms, the outer function can be represented as $$g(y) = y^7$$.

Step 3: Verify the composite function

To check if the composite function $$h(x)=g(f(x))$$ holds true, let's substitute $$f(x)$$ into $$g(y)$$. $$g(f(x)) = (7x^3 - 10x + 17)^7$$ Since this expression is the same as the given function $$h(x)$$, we have successfully found the two functions $$f(x)$$ and $$g(x)$$ such that $$h(x) = g(f(x))$$.

Step 4: Final Answer

The given function $$h(x)=\left(7 x^{3}-10 x+17\right)^{7}$$ can be written as the composite of the two functions $$f(x) = 7x^3 - 10x + 17$$ and $$g(x) = x^7$$, where $$h(x)=g(f(x))$$.

Key Concepts

Understanding Function Composition

In mathematics, the concept of function composition is fundamental and often links two or more functions to create a new function. To understand this, imagine a function as a machine that takes an input and transforms it into an output. When we talk about composing functions, it's akin to feeding the output of one machine into another to produce a final result.

Viewing the exercise given, the function \(h(x) = (7x^3 - 10x + 17)^7\) is a composite function because it can be built from two simpler functions. As seen in the steps, we determine the inner function first, which in this case is \(f(x) = 7x^3 - 10x + 17\), representing the cubic expression inside the parentheses. The outer function is \(g(y) = y^7\), representing the exponentiation.

By understanding that a composite function \(h(x)\) can be expressed as \(h(x) = g(f(x))\), we are interpreting that the output from \(f(x)\), which is the evaluated polynomial, becomes the input for \(g(y)\), which then exponentiates that result. Such understanding builds a foundation to tackle more complex functions by breaking them down into more manageable parts.

Exploring Polynomial Functions

What Defines a Polynomial Function?

In the realm of algebra, polynomial functions are some of the most commonly encountered types of functions. They are expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

A polynomial function like \(f(x) = 7x^3 - 10x + 17\), which we identified as the inner function in the given composite function, showcases several characteristics of polynomials:

• The highest power of the variable, known as the degree, in this case, is 3, which classifies it as a cubic polynomial.
• Each term consists of a coefficient (such as 7, -10, or 17) and the variable \(x\) raised to an exponent.
• Polynomial functions are continuous and differentiable, meaning they have no breaks, holes, or sharp corners in their graphs and you can find their slope at every point.

Working with Polynomial Functions

When you're faced with a polynomial function, understanding its structure is crucial. Knowing that polynomials are usually written in standard form with the highest degree term first, makes it easier to recognize patterns and anticipate graph shapes. Moreover, finding factors can help in determining the roots of the function – points where the graph intersects the x-axis.

Deciphering Exponential Functions

The Power of Exponential Functions

Exponential functions, such as the outer function in our exercise \(g(y) = y^7\), exhibit a form where a constant base is raised to a variable exponent. Unlike polynomial functions where the variable is the base, here it's the power, which leads to an entirely different behavior and set of properties:

• Exponential growth or decay depending on the base; with \(y^7\), we see exponential growth.
• A defining feature of exponential functions is that their rate of change increases or decreases multiplicatively, in contrast to polynomial functions where the rate of change is additive.
• The graph of an exponential function is not symmetrical and can vary widely based on the exponent and the value of the base.

Applications of Exponential Functions

Exponential functions find their place in a variety of real-world applications, including compound interest in finance, population growth models in biology, and the decay of radioactive substances in physics. When composing polynomial functions with exponential functions, as in the provided exercise, we tap into a rich field of mathematical constructs that can model complex phenomena with simplicity and elegance.