Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If \(f\) is a linear polynomial, then \(f \circ f\) is a quadratic polynomial. c. If \(f\) and \(g\) are polynomials, then the degrees of \(f \circ g\) and \(g\) of are equal. d. To graph \(g(x)=f(x+2),\) shift the graph of \(f\) 2 units to the right.

Short answer

Question: Determine the truth or falsehood of each statement. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If f is a linear polynomial, then f ∘ f is a quadratic polynomial. c. If f and g are polynomials, then the degrees of f ∘ g and g are equal. d. To graph g(x) = f(x + 2), shift the graph of f 2 units to the right.

Step-by-step solution

a. Polynomial vs Rational Functions

A polynomial is a function of the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where the coefficients \(a_i\) are real numbers and n is a non-negative integer. A rational function is a function of the form \(R(x) =\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\). For any polynomial \(P(x)\), we can write it as a rational function like this: \(R(x)=\frac{P(x)}{1}\), since dividing by 1 does not change the polynomial. However, not all rational functions are polynomials. For example, \(R(x)=\frac{1}{x}\) is a rational function, but not a polynomial as its denominator (\(x\)) is not a constant. Therefore, the statement "All polynomials are rational functions, but not all rational functions are polynomials." is true.

b. Composition of Linear Polynomials

A linear polynomial is a polynomial of degree 1, which has the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The composition of f with itself, denoted by \(f ∘ f\), is given as: \(f(f(x)) = f(mx + b)\). Now, substitute the expression for f(x) into this composition: \(f(f(x)) = m(mx + b) + b = m^2x + mb + b\). This is a quadratic polynomial, as the highest power of x is 2 (from the term \(m^2x\)). Therefore, the statement "If \(f\) is a linear polynomial, then \(f ∘ f\) is a quadratic polynomial." is true.

c. Degrees of Composition of Polynomials

Let \(f(x)\) and \(g(x)\) be polynomials with degrees m and n, respectively. Then, the composition \(f(g(x))\) is formed by replacing every occurrence of x in f(x) with the polynomial g(x). Since \(g(x)\) has degree n, the term with the highest degree in the composition will be \(x^n\) raised to the power of m, giving a term of degree m*n in \(f(g(x))\). So, the degree of \(f(g(x))\) is m*n, which is not necessarily equal to the degree of \(g(x)\). As a counterexample, let \(f(x) = x^2\) and \(g(x) = x^3\). The degrees of \(f(x)\) and \(g(x)\) are 2 and 3, respectively. The composition is \(f(g(x)) = (x^3)^2 = x^6\). The degree of \(f(g(x))\) is 6, which is not equal to the degree of \(g(x)\), which is 3. Therefore, the statement "If \(f\) and \(g\) are polynomials, then the degrees of \(f ∘ g\) and \(g\) are equal." is false.

d. Shifting the Graph

Given a function \(g(x) = f(x + 2)\), we need to determine what kind of transformation the graph of \(f(x)\) undergoes to become the graph of \(g(x)\). When we replace \(x\) with \((x+2)\), we are introducing a horizontal shift to the left or right. To see this, consider what happens to the points on the graph of \(f(x)\). Any point \((x, f(x))\) on the graph of \(f(x)\) will become a point \((x-2, f(x+2))\) on the graph of \(g(x)\). This means that the point moves 2 units to the left instead of moving to the right. Therefore, the statement "To graph \(g(x)=f(x+2),\) shift the graph of \(f\) 2 units to the right." is false. The correct statement is "To graph \(g(x)=f(x+2),\) shift the graph of \(f\) 2 units to the left."

Key concepts

Polynomials

Polynomials are a fundamental concept in calculus and algebra. These functions take the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where each \(a_i\) is a real number and \(n\) is a non-negative integer. Polynomials are characterized by their degree, which is the highest power of \(x\) with a non-zero coefficient.

Polynomials can be simple, like linear polynomials (e.g., \(f(x) = 2x + 3\)) or more complex, like quadratic or cubic polynomials. They exhibit smooth, continuous curves and are differentiable at every point, making them a key focus for study in calculus due to their predictable behavior.

One interesting property of polynomials is that they can always be expressed as rational functions. You can write any polynomial \(P(x)\) as a rational function \(\frac{P(x)}{1}\). However, this does not work the other way around as not all rational functions are polynomials.

Rational Functions

Rational functions are quotients of two polynomials. They are expressed in the form \(R(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)eq 0\). These functions offer more complex behaviors than simple polynomials because they can have discontinuities, such as asymptotes or holes in the graph where the denominator equals zero.

An essential aspect of rational functions is understanding their domain, which is all real numbers except where \(Q(x) = 0\). These points of discontinuity must be treated carefully when graphing or analyzing the function's behavior. Occasionally, you may encounter a rational function like \(\frac{1}{x}\), which is not a polynomial because the denominator isn't a constant.

Rational functions can model situations that standard polynomials cannot, such as rates and ratios, making them highly useful in various fields of science and engineering.

Graph Transformations

Graph transformations are changes applied to the graph of a function that alter its position, shape, or orientation. Common transformations include translations (shifts), reflections, stretches, and compressions. Understanding these transformations is crucial for graphing different functions efficiently and predicting their behavior.

One frequent transformation is horizontal shifting. If you have a function \(f(x)\) and you define a new function \(g(x) = f(x + c)\), you are effectively shifting the graph of \(f(x)\) horizontally. A function \(g(x) = f(x + 2)\) shifts the graph of \(f(x)\) 2 units to the left, contrary to the first impression that it moves to the right. This occurs due to the interchangeable nature of addition in the horizontal direction.

Vertical shifts are more intuitive, such as moving the graph up with \(f(x) + c\) or down with \(f(x) - c\). Properly understanding graph transformations allows students to manipulate and interpret graphs swiftly, making them an indispensable tool in calculus.