Classify each function as a power function, root function,...

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. $$\begin{array}{ll}{\text { (a) } y=\pi^{x}} & {\text { (b) } y=x^{\pi}} \\ {\text { (c) } y=x^{2}\left(2-x^{3}\right)} & {\text { (d) } y=\tan t-\cos t} \\ {\text { (e) } y=\frac{s}{1+s}} & {\text { (f) } y=\frac{\sqrt{x^{3}-1}}{1+\sqrt[3]{x}}}\end{array}$$

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

$$\begin{array}{ll}{\text { (a) } y=\pi^{x}} & {\text { (b) } y=x^{\pi}} \\ {\text { (c) } y=x^{2}\left(2-x^{3}\right)} & {\text { (d) } y=\tan t-\cos t} \\ {\text { (e) } y=\frac{s}{1+s}} & {\text { (f) } y=\frac{\sqrt{x^{3}-1}}{1+\sqrt[3]{x}}}\end{array}$$

Key Concepts

Power Function

A power function is any function that can be written in the form f(x) = x^a, where a is a constant exponent. It represents a scaling relationship between the variable and its output through a fixed power, and it is a fundamental concept in algebra and calculus, often used to model direct proportional relationships.

Exponential Function

An exponential function is characterized by a constant base raised to a variable exponent and is of the form f(x) = a^x, where a is a positive constant not equal to one. Exponential functions are significant in describing growth or decay processes and are ubiquitous in fields such as finance, physics, and biology.

Polynomial Function

A polynomial function is a sum of one or more power functions with non-negative integer exponents, typically expressed as f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0. This category of functions is important due to its relatively simple behavior and the ease with which it can be differentiated and integrated.

Degree of a Polynomial

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial with a non-zero coefficient. It is a key characteristic that influences the polynomial’s graph, the number of roots it can have, and its end-behavior.

Rational Function

A rational function is defined as the quotient of two polynomial functions, that is, f(x) = P(x)/Q(x), where both P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial. Rational functions often exhibit asymptotic behavior and discontinuities, making them useful in modeling situations where division of quantities is natural.

Algebraic Function

An algebraic function is any function that satisfies a polynomial equation whose coefficients are themselves polynomials in the variable. This broad category includes power functions, polynomials, roots (expressed as fractional exponents), and any function that can be constructed using a finite number of operations including addition, subtraction, multiplication, division, and root extraction.

Trigonometric Function

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. These functions are periodic and are essential in studying oscillatory phenomena, waves, and circular motion.

Logarithmic Function

A logarithmic function is the inverse of an exponential function, typically written as f(x) = log_a(x), where a is a positive constant different from one. Logarithmic functions are crucial for solving equations involving exponents and for modeling phenomena that scale multiplicatively, such as pH in chemistry or the Richter scale in seismology.

Root Function

A root function involves the extraction of a root from its argument, which can be rewritten as a power function with a fractional exponent (for example, the square root of x is x^(1/2)). Root functions are a specific case of power functions and are common in equations describing geometric relationships.

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