What processes are needed to solve for functions and their inverses?

Inverse functions are usually denoted as g(x) and h(x), where g(x) is the original equation and h(x) is the corresponding inverse function. You can determine if the equations are inverse of each other if a certain point (a,b) on g(x) is on h(x) as point (b,a). So, there is a pattern wherein the x and y coordinates interchange with each other. Also, you could see it visually when you graph both equations. They are mirror images of each other about line y=x or a 45° line.

One way to solve if the equations are inverse of each other is if they meet this condition:

g( h(x) ) = h( g(x) )

Suppose g(x) = 3x + 7. To find its inverse, change g(x) to y first so as not to confuse you:

y = 3x + 7

Next, interchange y and x variables

x = 3y + 7

Isolate y on one side and x on the other:

y = (x - 7)/3 = h(x)

That is the inverse equation for g(x) denoted as h(x). To check, let's verify the condition:

g((x - 7)/3) = h(3x + 7)

3(x - 7)/3) + 7 ? ((3x + 7 - 7)/3

x - 7 + 7 ? 3x/3

x = x

Therefore, g(x) and h(x) are indeed inverse functions of each other.

the answer is 24

step-by-step explanation: