Find a formula for the inverse of f(x) = sqrt*(4x-3) with x as an independent variable
- Recognize what is being asked: f^-1(x) - Rewrite the equation by replacing f(x) with y: y = sqrt*(4x-3) - Next, solve for x as a function of y meaning, isolate x on one side of the equation. This is the same thing as saying x = f^-1 (y) y = sqrt*(4x-3) y^2 = 4x-3 y^2 + 3 = 4x (y^2 + 3)/4 = x x = 1/4 (y^2 + 3) f^-1 (y) = 1/4 (y^2 + 3) - Since the question is asking for x as an independent variable, to get the inverse function of x as f^-1(x), the next step is to replace y with x like so: f^-1 (x) = 1/4 (x^2 + 3)
Write the electron configuration of the following: Zn2+
To answer this question, we must consult the spdf electron blocks on the periodic table. - Recognize that Zn is in the "3d" block of the periodic table, which means all sub-shells beforehand, are filled to maximum capacity (1s2 2s2 2p6 3s2 3p6 4s2) - Counting from left to right, Zn falls in the 10th column out of the 10 columns in the "3d" block, this means that Zn has 10 electrons in the "3d" sub-shell. - The Zn atom electron configuration can be written as follows: 1s2 2s2 2p6 3s2 3p6 4s2 3d10. - Zn2+ is the same thing as Zn stripped of 2 electrons (indicated by positive charge). - Electrons are removed from the higher energy orbital first, in this case 4s. According to Bohr's model n = 4 > n=3 in terms of energy level. - The Zn2+ electron configuration can therefore be written as follows: 1s2 2s2 2p6 3s2 3p6 3d10.
College Algebra Question: Given f(x) = 4x+2 and (f + g)(x) = 3 - 1/2x. What is the function of g?
There are a few moving parts in this problem. But fear not, let's take it step by step! - Recognize what the problem is asking for: g(x) - Recognize where to get g(x) from the two provided functions: (f + g)(x) = 3 - 1/2x. - Recognize that (f + g)(x) = 3 - 1/2x is a "compound function" of two separate functions and can be broken down following the following formula: (f + g) (x) = f(x) + g(x). - Rewrite problem as needed: (f + g)(x) = 3 - 1/2x -----> f(x) + g(x) = 3 - 1/2x. - Recognize what is given to find what is needed. Given: f(x) + g(x) = 3 - 1/2x Given: f(x) = 4x + 2 Need: g(x) - Recognize that g(x) can be isolated as the sole denoted function by substituting f(x) in f(x) + g(x) = 3 - 1/2x with the provided equation f(x) = 4x + 2 like so: (f(x)) + g(x) = 3 - 1/2x (4x + 2) + g(x) = 3 - 1/2x - Recognize that the equation can be solved for g(x) by collecting like-terms on the opposite side of the equal sign from g(x) like so: 4x + 2 + g(x) = 3 - 1/2x - 2 -2 _________________ 4x + g(x) = 1 - 1/2x - 4x - 4x __________________ g(x) = 1 - 9/2x **[-1/2x - 4x = -1/2 x - 8/2x = -9/2x]** or g(x) = 1-4.5 x <----------------ANSWER - Double check that the answer cannot be simplified further. Note: In the same manner, similar problems can be solved by recognizing how these "compound functions" are broken down. Important formlas are listed below: - Sum of f+g ----> (f + g) (x) = f(x) + g(x). - Difference of f - g ----> (f - g) (x) = f(x) - g(x). - Product of f X g ----> (f X g) (x) = f(x) X g(x). - Quotient of f/g ----> (f /g) (x) = f(x) / g(x)