Find the domain of existence for each of the following functions: a) 10x5 - 2x + (1)/(3) b) (x)/(x2 - 4) c) [4]x-10

Hey Candido, good to see you again. Here are the domains of existence for each expression: a) 10x^5 - 2x + (1)/(3) Step 1: Classify the expression. This is a polynomial function. Step 2: Determine the domain of existence. Polynomial functions are defined for all real numbers. D = (-, +) The domain is R. b) (x)/(x^2 - 4) Step 1: Classify the expression. This is a rational function. Step 2: Determine the conditions for existence. For a rational function, the denominator cannot be zero. x^2 - 4 ≠ 0 Step 3: Solve for x where the denominator is zero. x^2 - 4 = 0 (x-2)(x+2) = 0 x-2 = 0 or x+2 = 0 x = 2 or x = -2 Step 4: State the domain. The function is defined for all real numbers except x=2 and x=-2. D = \x R x ≠ -2 and x ≠ 2\ In interval notation: D = (-, -2) (-2, 2) (2, +) The domain is R \-2, 2\. c) [4]x-10 Step 1: Classify the expression. This is a radical function with an even index (fourth root). Step 2: Determine the conditions for existence. For an even root, the expression under the radical must be non-negative. x-10 0 Step 3: Solve the inequality. x 10 Step 4: State the domain. D = \x R x 10\ In interval notation: D = [10, +) The domain is [10, +). d) sqrt(x+1) Step 1: Classify the expression. This is a radical function with an even index (square root). Step 2: Determine the conditions for existence. For an even root, the expression under the radical must be non-negative. x+1 0 Step 3: Solve the inequality. x -1 Step 4: State the domain. D = \x R x -1\ In interval notation: D = [-1, +) The domain is [-1, +). Send me the next one 📸