Question

Find a polynomial \(f\) that satisfies the following properties. (Hint: Determine the degree of \(f ;\) then substitute a polynomial of that degree and solve for its coefficients.) $$f(f(x))=9 x-8$$

Short answer

Question: Determine a polynomial \(f(x)\) such that \(f(f(x))=9x-8\). Answer: The possible polynomials for \(f(x)\) are \(f(x) = 3x-2\) or \(f(x) = -3x+4\).

Step-by-step solution

Determine the degree of \(f\)

Observe that \(f(f(x))=9x-8\). Since this expression is a linear function of \(x\), it means that \(f(x)\) must be of degree 1, because applying a linear function twice also results in a linear function. Therefore, let \(f(x)=ax+b\), where \(a\) and \(b\) are constants.

Calculate \(f(f(x))\)

Now that we know \(f(x)=ax+b\), we need to find the value of \(f(f(x))\). First, we'll substitute \(x\) with \(f(x)\) in the expression of \(f(x)\). This gives us \(f(f(x))=a(ax+b)+b = a^2x+ab+b\).

Solve for the coefficients \(a\) and \(b\)

We are given that \(f(f(x)) = 9x-8\). Equating it with the expression we found in Step 2, $$a^2x+ab+b=9x-8$$ Now, comparing the coefficients of the linear terms and the constant terms, we get two simultaneous equations: $$a^2 = 9 \Rightarrow a = \pm3$$ $$ab+b = -8$$ Solving for \(b\) when \(a = 3\): $$3b+b=-8 \Rightarrow 4b=-8 \Rightarrow b= -2$$ Solving for \(b\) when \(a=-3\): $$-3b+b=-8 \Rightarrow -2b=-8 \Rightarrow b=4$$ We have two solutions for \(f(x)\): 1. \(f(x) = 3x-2\) 2. \(f(x) = -3x+4\) Either of these two polynomials satisfies the given conditions.

Key concepts

Degree of a Polynomial

When discussing polynomial functions, the term 'degree' is fundamental. It refers to the highest exponent of the variable in the polynomial’s expression. For instance, in the polynomial \(3x^4 - 5x^3 + 2x - 7\)\, the degree is 4 since the largest exponent is attached to the \(x^4\)\ term.

Understanding the degree is essential for predicting the behavior of the function, such as the number of possible roots and the general shape of its graph. In our previous exercise, \(f(f(x)) = 9x - 8\)\ is a linear function with a degree of 1, indicating its graph is a straight line. Consequently, \(f(x)\)\ must also be of degree 1, since the composition of two polynomials results in a polynomial whose degree is the product of the two original degrees. Therefore, if \(f(x)\)\ was anything other than a linear function (degree 1), the composition \(f(f(x))\)\ wouldn't result in a linear function.

Solving for Coefficients

To determine a polynomial function completely, we need to find its coefficients. This requires a set of conditions or equations that the polynomial must satisfy. In our exercise, \(f(f(x)) = 9x - 8\)\ provides such a condition. Once we've allowed for the degree, we articulate a general polynomial with undetermined coefficients. In this case, for a degree 1 polynomial, \(f(x) = ax + b\)\.

We then insert this polynomial into the given conditions, resulting in an equation or a system of equations that we can solve for the coefficients \(a\)\ and \(b\)\. Particularly, by equating the coefficients of the like terms of \(f(f(x))\)\ and the given polynomial equation \(9x - 8\)\, we're able to solve for the values of \(a\)\ and \(b\)\. This method is known as the method of coefficients, which is widely used in algebra to solve for unknowns.

Composition of Functions

Composition of functions involves creating a new function by applying one function to the result of another. Expressed symbolically, the composition of functions \(f\)\ and \(g\)\ is written as \(f(g(x))\)\, which reads as '\(f\)\ of \(g\)\ of \(x\)\'. This is an operation that takes a value \(x\)\, applies the function \(g\)\ to it, and then applies the function \(f\)\ to the result.

In the context of our exercise, we were tasked to find a function \(f\)\ that when composed with itself—meaning \(f(f(x))\)—yields the linear function \(9x - 8\)\. This is a special kind of composition because it involves the function with itself, which is often called an iterated function. It is crucial to first discern that the resulting composition must reflect the degree of the original function as exponentiated by itself; a vital insight for correctly approaching and solving for the initial function, \(f(x)\)\.