The onetoone functions g and h are defined as follows g 9 7

The one-to-one functions g and h are defined as follows. g = {(-9, -7), (-8, 2), (1, 7), (2, -8)} h (x) = 3x + 4 Find the following

Solution

The one-to-one functions g and h are defined as follows. g={(-9,-7), (-8,2), (7,1)(2,-8)} h(x)= 3x+4 Find the following. g-1(2)= \"g-1(2)\" just says \"Find the pair of coordinates that has 2 for its y-coordinate, and the answer is its x-coordinate\". So we look through those and find (-8,2) is the only one of those up there that has a 2 for it\'s y- coordinate, and so its x-coordinate is -8 and we write: g-1(4)=-8 h-1(x)= Start with h(x) = 3x+4 Change \"h(x): to \"y\" y = 3x+4 Interchange x and y: x = 3y+4 Solve for y: x-4 = 3y (x-4)/3 = y Change y to h^-1(x) h-1(x) = (x-4)/3 (h-1 o h)(-2)= That\'s the same as: h-1(h(-2)) = ? First find h(-2) h(x) = 3x+4 h(-2) = 3(-2)+4 h(-2) = -6+4 h(-2) = -2 Then h-1(h(-2)) = h-1(-2) = then find h-1 of -2 by plugging -2 in for x. h-1(x) = (x-4)/3 h-1(-2) = (-2-4)/3 h-1(7) = -6/3 h-1(7) = -2 h-1(h(-2)) = h-1(-2) = -2. So when a function is composed with its inverse, you get a function that gives you back the same number for y that you substituted for x. That\'s analogous to -2x1=-2, like multiplying by 1 or adding 0.