The function f is defined by f x = 4x2 − 12x + 13 for p < x < q, where p and q are constants. Thefunction g is defined by g x = 3x + 1 for x < 8.(b) Given that it is possible to form the composite function gf, find the least possible value of p andthe greatest possible value of q.
Question
The function f is defined by f�x = 4x2 − 12x + 13 for p < x < q, where p and q are constants. Thefunction g is defined by g�x = 3x + 1 for x < 8.(b) Given that it is possible to form the composite function gf, find the least possible value of p andthe greatest possible value of q.
Solution
To form the composite function gf, the range of f must be within the domain of g.
The domain of g is x < 8, so the range of f must be less than 8.
The function f is a quadratic function that opens upwards (since the coefficient of x^2 is positive). This means it has a minimum value.
The minimum value of a quadratic function given in the form f(x) = ax^2 + bx + c is at x = -b/2a.
For the function f, a = 4 and b = -12, so the minimum value is at x = -(-12)/(2*4) = 1.5.
The value of f at x = 1.5 is f(1.5) = 4*(1.5)^2 - 12*1.5 + 13 = 4.5.
Since 4.5 < 8, the range of f is within the domain of g, so it is possible to form the composite function gf.
However, for the composite function to be defined, we need to find the values of p and q such that p < x < q is within the domain of g.
Since the domain of g is x < 8, the greatest possible value of q is 8.
Since the minimum value of f is at x = 1.5 and we need p < x, the least possible value of p is any number less than 1.5.
So, the least possible value of p is less than 1.5 and the greatest possible value of q is 8.
Similar Questions
wo functions are shown below.f(x) = 12 • 2xg(x) = 5x + 2What is the largest integer value of x such that f(x) ≤ g(x)?
The functions f, g and h are defined byf :g:h:x xx xx x x6663 1211 12−+ ≠ −(a) Find (i) g( )6(ii) h −⎛⎝⎜ ⎞⎠⎟13 (2)(b) Express hf(x) in terms of x, simplifying your answer.(1)(c) Solve the equation(i) g(x) = 258(ii) gf(x) = x(7)..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Given the function( ) ( 3)( 2)y x x x= + − , for4 4x− . What is the value of x atwhich the function has a minimum?(A)32−(B)12−(C)12(D)32
and , so, the value of composite function f(g(2)) is ____.Question 3Answera.64b.74c.73d.63
Question No 54.Let f(x) and h(x) be functions defined as f(x) = 2x + 4 and h(x) = 2(2x – 2) – 2(x + 2) + 16. Find the largest value of 'x' such that f(x) = 4h(x).
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.