Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Classify each of the following as a PF (polynomial function), RF (rational function but not a polynomial function), or neither. (a) \(f(x)=3 x^{1 / 2}+1\) (b) \(f(x)=3\) (c) \(f(x)=3 x^{2}+2 x^{-1}\) (d) \(f(x)=\pi x^{3}-3 \pi\) (e) \(f(x)=\frac{1}{x+1}\) (f) \(f(x)=\frac{x+1}{\sqrt{x+3}}\)

Step-by-Step Solution

Answer

(a) Neither, (b) PF, (c) RF, (d) PF, (e) RF, (f) RF.

Step 1: Evaluate Expression (a)

The function given is \( f(x) = 3x^{1/2} + 1 \). A polynomial function consists of terms with non-negative integer exponents. The term \( x^{1/2} \) has a fractional exponent, thus it cannot be classified as a polynomial function. Since it is not a rational function (a quotient of polynomials) either, it is classified as 'neither'.

Step 2: Evaluate Expression (b)

The function given is \( f(x) = 3 \). It can be considered a polynomial function, as it can be written as \( 3x^0 \), where the exponent is a non-negative integer (0 in this case). Hence, it is classified as a PF (polynomial function).

Step 3: Evaluate Expression (c)

The function given is \( f(x) = 3x^2 + 2x^{-1} \). A polynomial function consists of terms with non-negative integer exponents. The term \( x^{-1} \) has a negative exponent, thus making it not a polynomial. This function can be seen as a quotient of polynomials (with \( 2x^{-1} \) representing \( \frac{2}{x} \)), classifying it as a rational function but not a polynomial: RF.

Step 4: Evaluate Expression (d)

The function given is \( f(x) = \pi x^3 - 3\pi \). All terms have exponents that are non-negative integers (3 and 0). Both coefficients, \( \pi \), are constants, which is standard in polynomial functions. Thus, this is classified as a PF (polynomial function).

Step 5: Evaluate Expression (e)

The function given is \( f(x) = \frac{1}{x + 1} \). This expression is clearly a quotient of two polynomials (1 is a polynomial of degree 0, and \( x+1 \) is a polynomial of degree 1), making it a rational function but not a polynomial function: RF.

Step 6: Evaluate Expression (f)

The function given is \( f(x) = \frac{x+1}{\sqrt{x+3}} \). The denominator contains a root (\( \sqrt{x+3} = (x+3)^{1/2} \)), indicating a non-polynomial term due to the fractional exponent. This makes the function a rational function but not a polynomial function: RF.

Key Concepts

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in mathematics. They consist of terms with variables raised to non-negative integer exponents. For example, expressions like \(5x^2 + 3x + 7\) or \(x^4 - 2x^2 + 1\) are polynomial functions. These functions have several key characteristics:

• **Non-negative integer exponents**: All exponents in a polynomial must be whole numbers (0, 1, 2, etc.).
• **Real-number coefficients**: The numbers before the variables, called coefficients, can be any real number, including fractions or irrational numbers like \(\pi\).
• **Continuous and smooth curves**: Graphs of polynomial functions are smooth and continuous, without any breaks or sharp corners.

In the above exercise, the expression \(f(x) = 3\) from Step 2 is a polynomial function because it can be written as \(3x^0\), and \(f(x) = \pi x^3 - 3\pi\) from Step 4 is also a polynomial due to its integer exponents of 3 and 0.

Rational Functions

Rational functions are a bit more complex than polynomial functions. They are defined as the quotient of two polynomial functions. In other words, a rational function is any function that can be expressed as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are both polynomials, and importantly \(Q(x)\) cannot be zero. Typical features of rational functions include:

• **Quotient of polynomials**: The function is the result of dividing one polynomial by another.
• **Open at discontinuities**: The graph of a rational function may have asymptotes or holes where the denominator equals zero.
• **Flexible shape**: Rational functions can take various forms, reflecting their complexity compared to polynomials.

In the exercise, \(f(x) = 3x^2 + 2x^{-1}\) from Step 3 is not a polynomial but is classified as a rational function because \(2x^{-1}\) can be rewritten as \(\frac{2}{x}\). Similarly, \(f(x) = \frac{1}{x+1}\) in Step 5 and \(f(x) = \frac{x+1}{\sqrt{x+3}}\) in Step 6 are both rational functions due to their form as quotients.

Non-Polynomial Functions

Non-polynomial functions cover a wide range of functions that do not fit the criteria for polynomials. These include functions with fractional, negative, or irrational exponents and those involving operations such as square roots or trigonometric functions. A few characteristics of non-polynomial functions are:

• **Fractional or negative exponents**: Unlike polynomials, non-polynomial functions may have terms like \(x^{1/2}\) or \(x^{-1}\).
• **Inclusion of diverse functions**: This category includes exponential, logarithmic, and trigonometric functions, among others.
• **No strict graph features**: Non-polynomial functions do not follow the smooth curve rule of polynomials.

In our problem set, the expression \(f(x) = 3x^{1/2} + 1\) from Step 1 is considered neither a polynomial nor a rational function due to the fractional exponent \(x^{1/2}\). This defines it as a non-polynomial function, showcasing the variety within this classification.