{"id":147,"date":"2023-09-20T22:47:52","date_gmt":"2023-09-20T22:47:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/combining-functions\/"},"modified":"2025-08-17T15:39:28","modified_gmt":"2025-08-17T15:39:28","slug":"review-of-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/review-of-functions-learn-it-5\/","title":{"raw":"Review of Functions: Learn it 5","rendered":"Review of Functions: Learn it 5"},"content":{"raw":"

Combining and Composing Functions<\/h2>\r\n

In mathematics, we often build upon basic functions by combining them with operations such as addition and multiplication or by creating composite functions. This process forms new functions that can better describe and analyze complex relationships.<\/p>\r\n

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Combining Functions with Mathematical Operators<\/h3>\r\n

To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions [latex]f[\/latex] and [latex]g,[\/latex] we can define four new functions:<\/p>\r\n

[latex]\\begin{array}{cccc}(f+g)(x)=f(x)+g(x)\\hfill & & & \\text{Sum}\\hfill \\\\ (f-g)(x)=f(x)-g(x)\\hfill & & & \\text{Difference}\\hfill \\\\ (f\u00b7g)(x)=f(x)g(x)\\hfill & & & \\text{Product}\\hfill \\\\ \\Big(\\frac{f}{g}\\Big)(x)=\\frac{f(x)}{g(x)} \\, \\text{for} \\, g(x)\\ne 0\\hfill & & & \\text{Quotient}\\hfill \\end{array}[\/latex]<\/div>\r\n
\u00a0<\/div>\r\n
Often these functions have more than one term, so when you perform operations on them you will need to remember how to work with polynomials.<\/div>\r\n
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When given multiple polynomials, you can simplify expressions by adding or subtracting them, ensuring you combine like terms and rearrange the resulting polynomial into standard form, which is organized by descending powers.\u00a0 Multiplying binomials requires a different approach. Use the FOIL method to multiply the first, outer, inner, and last terms, and then combine like terms in the resulting expression.<\/p>\r\n

Here's a concise recap of both processes:<\/p>\r\n

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    Adding\/Subtracting Polynomials:<\/strong><\/p>\r\n

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    1. Combine like terms.<\/li>\r\n\t
    2. When subtracting, distribute the negative sign across the second polynomial.<\/li>\r\n\t
    3. Rearrange and combine terms into standard form.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n

      Be particularly cautious when subtracting polynomials to distribute the negative sign correctly.<\/p>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"487\"]\"Two Diagram of FOIL terms[\/caption]\r\n\r\n

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        Multiplying Binomials (FOIL):<\/strong><\/p>\r\n

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        1. Multiply the first terms of each binomial.<\/li>\r\n\t
        2. Multiply the outer terms.<\/li>\r\n\t
        3. Multiply the inner terms.<\/li>\r\n\t
        4. Multiply the last terms.<\/li>\r\n\t
        5. Combine like terms and simplify the product.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/div>\r\n
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          Given the functions [latex]f(x)=2x-3[\/latex] and [latex]g(x)=x^2-1[\/latex], find each of the following functions and state its domain.<\/p>\r\n

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          1. [latex](f+g)(x)[\/latex]<\/li>\r\n\t
          2. [latex](f-g)(x)[\/latex]<\/li>\r\n\t
          3. [latex](f\u00b7g)(x)[\/latex]<\/li>\r\n\t
          4. [latex]\\Big(\\frac{f}{g}\\Big)(x)[\/latex]<\/li>\r\n<\/ol>\r\n

            [reveal-answer q=\"fs-id1170572228247\"]Show Solution[\/reveal-answer]
            \r\n[hidden-answer a=\"fs-id1170572228247\"]<\/p>\r\n

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            1. [latex](f+g)(x)=(2x-3)+(x^2-1)=x^2+2x-4[\/latex]. The domain of this function is the interval [latex](\u2212\\infty ,\\infty )[\/latex].<\/li>\r\n\t
            2. [latex](f-g)(x)=(2x-3)-(x^2-1)=\u2212x^2+2x-2[\/latex]. The domain of this function is the interval [latex](\u2212\\infty ,\\infty)[\/latex].<\/li>\r\n\t
            3. [latex](f\u00b7g)(x)=(2x-3)(x^2-1)=2x^3-3x^2-2x+3[\/latex]. The domain of this function is the interval [latex](\u2212\\infty ,\\infty )[\/latex].<\/li>\r\n\t
            4. [latex]\\Big(\\frac{f}{g}\\Big)(x)=\\dfrac{2x-3}{x^2-1}[\/latex]. The domain of this function is [latex]\\{x|x\\ne \\text{\u00b1}1\\}[\/latex].<\/li>\r\n<\/ol>\r\n\r\nWatch the following video to see the worked solution to this example.