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)

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
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).

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For example, if f(x) and g(x) are two functions, then f+g is defined as (f+g)(x) = f(x) + g(x). Understanding how to perform these operations is essential for solving the given problem.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero. In this case, identifying the domain for f(x) and g(x) requires finding the values of x that would lead to division by zero.

Rational Functions

Rational functions are ratios of two polynomials. They can exhibit unique behaviors such as asymptotes and discontinuities, particularly where the denominator equals zero. In the given functions f(x) and g(x), recognizing their rational nature is crucial for understanding their domains and how to perform operations on them.