Calculating Inverse Functions for Hourly Salary and Production Units

[Unichoran]

Homework Statement

Hello,I have some problems with my Pre-Calculus homework. The task is:
You get paid 8$ per hour plus 0.85$ per unit you produced.
1.Set up an equation for it.
2.Find the inverse function.

3.What does each variable in the inverse function mean?

Homework Equations

See below

The Attempt at a Solution

So I set up the equation for the salary per hour:
x=8+0.85*u
u is the amount of produced units,x the hourly salary.But I absolutely don't know how to get the inverse function of it.Could someone please help me with that?

 

[andrewkirk]

Just make u the subject of the equation.

 

[Ray Vickson]

Homework Statement

Hello,I have some problems with my Pre-Calculus homework. The task is:
You get paid 8$ per hour plus 0.85$ per unit you produced.
1.Set up an equation for it.
2.Find the inverse function.

3.What does each variable in the inverse function mean?

Homework Equations

See below

The Attempt at a Solution

So I set up the equation for the salary per hour:
x=8+0.85*u
u is the amount of produced units,x the hourly salary.But I absolutely don't know how to get the inverse function of it.Could someone please help me with that?

Solve for u in terms of x---that is exactly what "inverse function" means. Alternatively, plot x vs u (with horizontal u-axis and vertical x-axis; now turn your graph paper through 90 degrees, so your x-axis is now horizontal and your u-axis is vertical. Now your graph is that of the inverse function!

 

[Unichoran]

Solve for u in terms of x---that is exactly what "inverse function" means. Alternatively, plot x vs u (with horizontal u-axis and vertical x-axis; now turn your graph paper through 90 degrees, so your x-axis is now horizontal and your u-axis is vertical. Now your graph is that of the inverse function!

So the inverse would be u=8+0.85*x?

 

[Mark44]

So the inverse would be u=8+0.85*x?

No.
All you did was switch x and u.
You started with x = 8 + 0.85u.
Solve this equation for u in terms of x. That will give you the inverse.

 

[Mark44]

Here's an example that might be helpful, with y = f(x) = $(x - 1)^3$
As it turns out, this function is one-to-one, so it has an inverse that is itself a function.

To find the inverse, we want to solve the equation above for x in terms of (as a function of) y.

$y = (x - 1)^3$
$\iff y^{1/3} = x - 1$
$\iff y^{1/3} + 1 = x$
So, x = y^1/3 + 1 = f^-1(y)

The two equations y = (x - 1)³ and x = y^1/3 + 1 are equivalent, which means that every pair of numbers (x, y) that lies on the graph of the first equation also lies on the graph of the second equation. Really, we have only one graph.

As a quick sanity check, if x = 1, then y = 0 in the first equation. and if y = 0, then x = 1 in the second equation. This confirms that (1, 0) is a solution to both equations. In fact, if a given ordered pair satisfies one equation, it will also satisfy the other equation.

Finally, if the problem asks for the formula for the inverse as a function of x, we can write y = x^1/3 + 1 = f^-1(x). This is where the swapping of x and y occurs. In my opinion, though, this last process is the least important and least useful, but it's the easiest, so beginning students do this step and nothing else.

 

[Ray Vickson]

So the inverse would be u=8+0.85*x?

Did you check whether the formula u = 8 + .85*x satisfies the equation x = 8 + .85*u? In other words, did you solve the equation x = 8 + .85 u for u in terms of x, and then check your solution?