1) For f(x) = 3x + 9, we can rewrite as y = 3x + 9 and the reverse the x and y, which yields x = 3y + 9. If we solve for y, we'll have the inverse function.
x = 3y + 9
x - 9 = 3y
y = 1/3x - 3
2) I'm not sure what the question is here. The statement f(x) = x is the same as saying y = x. While y = x does not have a vertex as it is a line, 0,0 is a point on the line.
To find the vertex of g(x)=3|x-4|+1: we can find the vertex by identifying the smallest possible value we can get from under the absolute value sign, which will be 0 since an absolute value can't have a negative result. We get 0 under the absolute value when x = 4. That leaves us with y = 3 x 0 + 1, y = 1. So the vertex will be 4,1. Any larger or smaller value for x will lead to a larger y value.
3) f(x)=3(1/2)^x. The domain refers to all x values we are allowed to plug into the function. Since x could be negative, positive, or 0 in this case, the domain is all real numbers. The range refers to all resulting y values that are possible. The larger x gets, the smaller the result of 1/2 to the x power which means that as x approaches infinity, y approaches 0. 0 is the lower limit of the range since no exponent can make a positive expression negative. There is no upper limit, however. If we plug in negative values for x with a large absolute value, we'll get large results for y. So the domain is all real numbers, the range is y > 0.