{"id":153,"date":"2023-09-20T22:47:54","date_gmt":"2023-09-20T22:47:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/more-function-types\/"},"modified":"2024-08-31T02:14:22","modified_gmt":"2024-08-31T02:14:22","slug":"basic-classes-of-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-classes-of-functions-learn-it-5\/","title":{"raw":"Basic Classes of Functions: Learn It 5","rendered":"Basic Classes of Functions: Learn It 5"},"content":{"raw":"

Algebraic Functions<\/h2>\r\n

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function<\/strong> is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.<\/p>\r\n

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function<\/strong> is any function of the form [latex]f(x)=p(x)\/q(x)[\/latex], where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials. The following are some examples of rational functions.<\/p>\r\n

[latex]f(x)=\\dfrac{3x-1}{5x+2}[\/latex]\u00a0 and\u00a0 \u00a0[latex]g(x)=\\dfrac{4}{x^2+1}[\/latex]<\/div>\r\n

A root function<\/strong> is a power function of the form [latex]f(x)=x^{1\/n}[\/latex], where [latex]n[\/latex] is a positive integer greater than one. For example, [latex]f(x)=x^{1\/2}=\\sqrt{x}[\/latex] is the square-root function and [latex]g(x)=x^{1\/3}=\\sqrt[3]{x}[\/latex] is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, [latex]f(x)=\\sqrt{4-x^2}[\/latex] is an algebraic function.<\/p>\r\n

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algebraic functions<\/h3>\r\n

Algebraic functions are mathematical expressions combining constants and variables through operations like addition, multiplication, division, and taking roots. They encompass both rational functions, ratios of polynomials, and root functions, involving nth roots of the variable. Th<\/p>\r\n<\/section>\r\n

Transcendental Functions<\/h2>\r\n

Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions<\/strong> because they are said to \u201ctranscend,\u201d or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions.<\/p>\r\n

A trigonometric function relates the ratios of two sides of a right triangle. They are [latex]\\sin x,\\, \\cos x, \\, \\tan x, \\, \\cot x,\\, \\sec x[\/latex], and [latex]\\csc x[\/latex].\u00a0<\/p>\r\n

An exponential function is a function of the form [latex]f(x)=b^x[\/latex], where the base [latex]b>0, \\, b \\ne 1[\/latex].<\/p>\r\n

A logarithmic function<\/strong> is a function of the form [latex]f(x)=\\log_b(x)[\/latex] for some constant [latex]b>0, \\, b \\ne 1[\/latex], where [latex]\\log_b(x)=y[\/latex] if and only if [latex]b^y=x[\/latex].\u00a0<\/p>\r\n

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transcendental functions<\/h3>\r\n

Transcendental functions, including trigonometric, exponential, and logarithmic functions, are those which cannot be defined by a finite number of algebraic operations.<\/p>\r\n<\/section>\r\n

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Classify each of the following functions, as algebraic or transcendental.<\/p>\r\n

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  1. [latex]f(x)= \\dfrac{\\sqrt{x^3+1}}{4x+2}[\/latex]<\/li>\r\n\t
  2. [latex]f(x)=2^{x^2}[\/latex]<\/li>\r\n\t
  3. [latex]f(x)=\\sin (2x)[\/latex]<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"fs-id1170573420590\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"fs-id1170573420590\"]<\/p>\r\n

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    1. Since this function involves basic algebraic operations only, it is an algebraic function.<\/li>\r\n\t
    2. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)<\/li>\r\n\t
    3. As in part (b), this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.<\/li>\r\n<\/ol>\r\n\r\nWatch the following video to see the worked solution to this example.