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An introduction to the given problem is given below.This problem involves finding the domain of two functions and computing the composition of the two functions. It also requires finding the points of intersection between two given curves.

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An introduction to the given problem is given below.This problem involves finding the domain of two functions and computing the composition of the two functions. It also requires finding the points of intersection between two given curves. i) To find the domain of a function, we need to identify the values of x for which the function is defined. For f(x) = x^2 - 4x + 4, we note that it is a polynomial function, and all polynomial functions are defined for all real numbers. Therefore, the domain of f(x) is the set of all real numbers, which we can represent as (-∞, ∞). For g(x) = 1/√(x-1), we need to consider the denominator of the function. The square root of a negative number is not a real number, and we cannot divide by zero. Therefore, the domain of g(x) is all real numbers greater than 1, which we can represent as .ii) To compute (g o f)(x), we need to substitute f(x) into g(x) wherever we see x in g(x). Therefore, we have: (g o f)(x) = g(f(x)) = 1/(√f(x)-1) = 1/(√(x^2-4x+4)-1) = 1/(|x-2|-1) , where x≥2

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An introduction to the given problem is given below.This problem involves finding the domain of two functions and computing the composition of the two functions. It also requires finding the points of intersection between two given curves. i) To find the domain of a function, we need to identify the values of x for which the function is defined. For f(x) = x^2 - 4x + 4, we note that it is a polynomial function, and all polynomial functions are defined for all real numbers. Therefore, the domain of f(x) is the set of all real numbers, which we can represent as (-∞, ∞). For g(x) = 1/√(x-1), we need to consider the denominator of the function. The square root of a negative number is not a real number, and we cannot divide by zero. Therefore, the domain of g(x) is all real numbers greater than 1, which we can represent as .ii) To compute (g o f)(x), we need to substitute f(x) into g(x) wherever we see x in g(x). Therefore, we have: (g o f)(x) = g(f(x)) = 1/(√f(x)-1) = 1/(√(x^2-4x+4)-1) = 1/(|x-2|-1) , where x≥2 To find the points of intersection of the two curves, we need to set their equations equal to each other: 5x^2 - 3x - 2 = 2x - 1 Simplifying this equation, we get: 5x^2 - 5x - 1 = 0 We can solve this quadratic equation by using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a where a = 5, b = -5, and c = -1. Plugging in these values, we get: x = (-(-5) ± sqrt((-5)^2 - 4(5)(-1))) / 2(5) x = (5 ± sqrt(45)) / 10 Simplifying this expression, we get: x = (1 ± sqrt(5)) / 2 So the two points of intersection are: ( (1 + sqrt(5)) / 2, 2(1 + sqrt(5)) / 2 - 1) and ( (1 - sqrt(5)) / 2, 2(1 - sqrt(5)) / 2 - 1) or approximately: (1.618, 2.236) and (-0.618, -2.236) Explanation: a) i)The domain of f(x) is all real numbers, because the expression x^2-4x+4 is defined for all real values of x. The domain of g(x) is x>1, because the expression √x-1 is only defined for x greater than or equal to 1, and dividing by zero is undefined. ii)(g o f)(x) = g(f(x)) = g(x^2-4x+4) = 1/√(x^2-4x+4)-1. b) To find the points of intersection of the curves y=5x^2-3x-2 and y=2x-1, we need to set the two equations equal to each other and solve for x. We can then substitute the values of x back into one of the equations to find the corresponding y-values. The solution will give us the coordinates of the points of intersection.Thank you

Q−1: [6+6 marks] a) ​​​​​​​Given the functions 〖f(x)=x〗^2-4x+4 and g(x)=1/(√x-1) i) Find the domain of f(x) and g(x). ii) Compute (g o f)(x). b) Find the points of intersection of the curves y=5x^2-3x-2 and y=2x-1

Public Answer

Q−1: [6+6 marks] a)
Given the functions 〖f(x)=x〗^2-4x+4 and g(x)=1/(√x-1)
i) Find the domain of f(x) and g(x).
ii) Compute (g o f)(x).
b) Find the points of intersection of the curves y=5x^2-3x-2 and y=2x-1