Question

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)

Accepted Answer

Step 1: Understand the problem. You are tasked with finding the sum (ƒ+g), difference (ƒ−g), product (ƒg), and quotient (ƒ/g) of the two functions ƒ(x) = (3x+1)/(x² - 25) and g(x) = (2x−4)/(x² - 25). Additionally, you need to determine the domain for each resulting function. Step 2: Recall that the domain of a rational function is all real numbers except where the denominator equals zero. For both ƒ(x) and g(x), the denominator is x² - 25. Solve x² - 25 = 0 to find the restricted values of x. Factorize as (x - 5)(x + 5) = 0, so x = 5 and x = -5 are excluded from the domain. Step 3: To find ƒ+g, add the two functions: ƒ(x) + g(x) = [(3x+1)/(x² - 25)] + [(2x−4)/(x² - 25)]. Combine the numerators over the common denominator: [(3x+1) + (2x−4)] / (x² - 25). Simplify the numerator to get (5x−3)/(x² - 25). The domain excludes x = 5 and x = -5. Step 4: To find ƒ−g, subtract the two functions: ƒ(x) − g(x) = [(3x+1)/(x² - 25)] − [(2x−4)/(x² - 25)]. Combine the numerators over the common denominator: [(3x+1) − (2x−4)] / (x² - 25). Simplify the numerator to get (x+5)/(x² - 25). The domain remains the same: x ≠ 5 and x ≠ -5. Step 5: To find ƒg and ƒ/g, multiply and divide the functions respectively. For ƒg, multiply the numerators and denominators: [(3x+1)(2x−4)] / [(x² - 25)(x² - 25)]. For ƒ/g, divide the numerators and multiply by the reciprocal of the denominator: [(3x+1)/(x² - 25)] ÷ [(2x−4)/(x² - 25)] = [(3x+1)/(2x−4)]. For both cases, the domain excludes x = 5, x = -5, and for ƒ/g, also x = 2 (since 2x−4 = 0).

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)

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주요 개념

Function Operations

Domain of a Function

Rational Functions