Functions

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Step 1: Check each element of A. 1 maps to a. 2 maps to c. 3 maps to b. Step 2: Every element of the domain has exactly one image. Answer: R is a function.

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Step 1: Look at element 1. It has two images: a and b. Step 2: A function cannot give two images to the same domain element. Answer: R is not a function.

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Step 1: Domain is the set of first elements. Domain = {1, 2, 3} Step 2: Co-domain is the given target set. Co-domain = {5, 6, 7, 8} Step 3: Range is the set of actual images. Range = {5, 6, 7} Answer:…

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Use the formula: Number of functions from A to B = q^p. Here, p = 3 and q = 4. Step 1: Substitute values. 4^3 = 64 Answer: 64 functions.

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Step 1: Denominator cannot be zero. x^2 - x - 6 = (x - 3)(x + 2) Step 2: Set each factor not equal to zero. x - 3 ≠ 0 and x + 2 ≠ 0 So x ≠ 3 and x ≠ -2 Step 3: Write the domain. Answer: Domain = R - {…

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Step 1: The quantity inside the square root must be non-negative. x - 3 ≥ 0 Step 2: Solve the inequality. x ≥ 3 Answer: Domain = [3, ∞).

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Step 1: Square of any real number is never negative. x^2 ≥ 0 Step 2: The smallest value occurs at x = 0. f(0) = 0 Step 3: Larger values are possible for other real x. Answer: Range = [0, ∞).

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For y = x^2: Each x has exactly one y value. So it represents a function. For y^2 = x: When x = 1, y = 1 or y = -1. So one x gives two y values. Answer: y = x^2 represents a function, but y^2 = x does…