Find the domain of the following functions: 1. f(x) = x/(x2+4) 2. f(x) = sqrt(x-4) 3. f(x) = sqrt(x2-16) 4. f(x) = sqrt(3-x)

Here are the solutions to the problems: Domain Problems 1. To determine the domain of (x)/(x^2+4), we need to ensure the denominator is not zero. Step 1: Set the denominator equal to zero. x^2+4 = 0 Step 2: Solve for x. x^2 = -4 There are no real values of x for which x^2 = -4. Thus, the denominator is never zero. The domain is all real numbers. (-, ) 2. To find the domain of sqrt(x-4), the expression under the square root must be non-negative. Step 1: Set the expression under the square root greater than or equal to zero. x-4 0 Step 2: Solve for x. x 4 The domain is all real numbers greater than or equal to 4. [4, ) 3. To determine the domain of sqrt(x^2-16), the expression under the square root must be non-negative. Step 1: Set the expression under the square root greater than or equal to zero. x^2-16 0 Step 2: Factor the inequality. (x-4)(x+4) 0 Step 3: Find the critical points where the expression equals zero, which are x=4 and x=-4. Test intervals to find where the inequality holds true. The expression is non-negative when x -4 or x 4. (-, -4] [4, ) 4. To find the domain of sqrt(3-x), the expression under the square root must be non-negative. Step 1: Set the expression under the square root greater than or equal to zero. 3-x 0 Step 2: Solve for x. 3 x x 3 The domain is all real numbers less than or equal to 3. (-, 3] 5. If f(x) = sqrt(4-x^2), to find the domain, the expression under the square root must be non-negative. Step 1: Set the expression under the square root greater than or equal to zero. 4-x^2 0 Step 2: Rearrange the inequality. x^2 4 Step 3: Take the square root of both sides, remembering to consider both positive and negative roots. -2 x 2 The domain is all real numbers between -2 and 2, inclusive. [-2, 2] 6. To determine the domain and range of |x|: Step 1: For the domain, consider all possible real values for x. The absolute value function is defined for all real numbers. Step 2: For the range, consider the output values of the absolute value function. The absolute value of any real number is always non-negative. Domain: (-, ) Range: [0, ) Limits of a Function 1. Evaluate _x 0 (7^x-3)/(5^x+3) Step 1: Substitute x=0 into the expression. _x 0 (7^x-3)/(5^x+3) = (7^0-3)/(5^0+3) Step 2: Simplify the expression. = (1-3)/(1+3) = (-2)/(4) = -(1)/(2) -(1)/(2) 2. Evaluate _x (x^3-x^2+4x)/(2x^3+2x^2-7) Step 1: Identify the highest power of x in the numerator and denominator. Both are x^3. Step 2: The limit as x of a rational function where the degrees of the numerator and denominator are equal is the ratio of their leading coefficients. _x (x^3-x^2+4x)/(2x^3+2x^2-7) = (1)/(2) (1)/(2) 3. Evaluate _x (x)/(x-6) Step 1: Identify the highest power of x in the numerator and denominator. Both are x^1. Step 2: The limit as x of a rational function where the degrees of the numerator and denominator are equal is the ratio of their leading coefficients. _x (x)/(x-6) = (1)/(1) = 1 1 4. Compute _x 6 (x-6)/(x^2-36) Step 1: Direct substitution results in (0)/(0), an indeterminate form. Factor the denominator. _x 6 (x-6)/((x-6)(x+6)) Step 2: Cancel out the common factor (x-6). = _x 6 (1)/(x+6) Step 3: Substitute x=6 into the simplified expression. = (1)/(6+6) = (1)/(12) (1)/(12) 5. Evaluate _x 6 (x-6)/(x^2+36) Step 1: Substitute x=6 into the expression. _x 6 (x-6)/(x^2+36) = (6-6)/(6^2+36) Step 2: Simplify the expression. = (0)/(36+36) = (0)/(72) = 0 0 6. Evaluate _x 1 (x^2-2x+1)/(x-1) Step 1: Direct substitution results in (0)/(0), an indeterminate form. Factor the numerator. _x 1 ((x-1)^2)/(x-1) Step 2: Cancel out the common factor (x-1). = _x 1 (x-1) Step 3: Substitute x=1 into the simplified expression. = 1-1 = 0 0 7. Evaluate _x 2 (x-2)/(x^2+2x-6) Step 1: Substitute x=2 into the expression. _x 2 (x-2)/(x^2+2x-6) = (2-2)/(2^2+2(2)-6) Step 2: Simplify the expression. = (0)/(4+4-6) = (0)/(2) = 0 0 8. Evaluate _x 4 (x^2-16)/(x-4) Step 1: Direct substitution results in (0)/(0), an indeterminate form. Factor the numerator. _x 4 ((x-4)(x+4))/(x-4) Step 2: Cancel out the common factor (x-4). = _x