Math 111 Worksheet 1 Solutions Show All Work

Question

Math 111 Worksheet 1 Solutions Show All Work Solut Evaluate and explain the domain of the function f(x) = √(x + 5) / (x - 2), using interval notation. Provide the rationale for your solution based on the constraints imposed by the radicand and the denominator. The function f(x) involves a square root in the numerator and a denominator that cannot be zero. To find the domain, we need to ensure that the radicand of the square root is greater than or equal to zero, because square roots of negative numbers are undefined in real numbers. Additionally, the denominator must not be zero to avoid division by zero errors. The numerator involves √(x + 5). Thus, the radicand x + 5 must be ≥ 0, so: x + 5 ≥ 0 → x ≥ -5 The denominator is (x - 2), which cannot be zero, so: x - 2 ≠ 0 → x ≠ 2 Combining these constraints, the domain is all real numbers greater than or equal to -5, except x ≠ 2. In interval notation, this is expressed as: [ -5, 2 ) ∪ ( 2, ∞ ) Algebraic evaluation of (f ◦ g)(x) as a single fraction Given the functions: f(x) = x + 3 g(x) = x - 1 We need to evaluate (f ◦ g)(x) = f(g(x)). First, find g(x): g(x) = x - 1 Next, evaluate f at g(x): f(g(x)) = f(x - 1) = (x - 1) + 3 = x + 2 Thus, (f ◦ g)(x) = x + 2, which is already in simplified form. Modeling projectile height and analysis The problem provides the maximum height of a projectile, which occurs at t = 0, with h(0) = 400 feet. The height function without air resistance is modeled as: h(t) = -16t² + v₀t + h₀ Since the maximum height is at t=0, and the maximum height is 400, the model simplifies to: h(t) = -16t² + 400 This quadratic function indicates that at t=0, the height is 400 feet, and it opens downward (negative coefficient for t²) consistent with gravity's effect. In physical terms, ignoring air resistance, the height of the projectile during its ascent and descent is described accurately by this parabola. The maximum height is 400 feet at t=0, representing the peak point in the projectile's trajectory. The

Dr. Jack HW Helper · Accepted Answer

Math 111 Worksheet 1 Solutions Show All Work Solut Evaluate and explain the domain of the function f(x) = √(x + 5) / (x - 2), using interval notation. Provide the rationale for your solution based on the constraints imposed by the radicand and the denominator. The function f(x) involves a square root in the numerator and a denominator that cannot be zero. To find the domain, we need to ensure that the radicand of the square root is greater than or equal to zero, because square roots of negative numbers are undefined in real numbers. Additionally, the denominator must not be zero to avoid division by zero errors. The numerator involves √(x + 5). Thus, the radicand x + 5 must be ≥ 0, so: x + 5 ≥ 0 → x ≥ -5 The denominator is (x - 2), which cannot be zero, so: x - 2 ≠ 0 → x ≠ 2 Combining these constraints, the domain is all real numbers greater than or equal to -5, except x ≠ 2. In interval notation, this is expressed as: [ -5, 2 ) ∪ ( 2, ∞ ) Algebraic evaluation of (f ◦ g)(x) as a single fraction Given the functions: f(x) = x + 3 g(x) = x - 1 We need to evaluate (f ◦ g)(x) = f(g(x)). First, find g(x): g(x) = x - 1 Next, evaluate f at g(x): f(g(x)) = f(x - 1) = (x - 1) + 3 = x + 2 Thus, (f ◦ g)(x) = x + 2, which is already in simplified form. Modeling projectile height and analysis The problem provides the maximum height of a projectile, which occurs at t = 0, with h(0) = 400 feet. The height function without air resistance is modeled as: h(t) = -16t² + v₀t + h₀ Since the maximum height is at t=0, and the maximum height is 400, the model simplifies to: h(t) = -16t² + 400 This quadratic function indicates that at t=0, the height is 400 feet, and it opens downward (negative coefficient for t²) consistent with gravity's effect. In physical terms, ignoring air resistance, the height of the projectile during its ascent and descent is described accurately by this parabola. The maximum height is 400 feet at t=0, representing the peak point in the projectile's trajectory. The

Math 111 Worksheet 1 Solutions Show All Work

Math 111 Worksheet 1 Solutions Show All Work Solut

Algebraic evaluation of (f ◦ g)(x) as a single fraction

Modeling projectile height and analysis

Additional Rocket and Radiation Source Model

Summary and practical interpretation:

References