# Calculating Inverse Functions for Hourly Salary and Production Units

• Unichoran
In summary, the conversation involves a student seeking help with their Pre-Calculus homework, specifically setting up an equation for hourly salary and finding the inverse function. The student is advised to solve for u in terms of x and given an example of finding an inverse function. The correct inverse function is u = (x-8)/0.85.
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?

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?

Just make u the subject of the equation.

Unichoran said:

## 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?

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
Ray Vickson said:
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?

Unichoran said:
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.

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 = y1/3 + 1 = f-1(y)

The two equations y = (x - 1)3 and x = y1/3 + 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 = x1/3 + 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.

Unichoran said:
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?

## 1. What is an inverse function?

An inverse function is a function that "undoes" the effect of another function. In other words, if f(x) is a function, its inverse function, denoted as f-1(x), will undo the action of f(x) and return x. This means that f(f-1(x)) = x and f-1(f(x)) = x.

## 2. How do you find an inverse function?

To find an inverse function, you need to follow these steps:
2. Replace f(x) with y.
3. Switch the x and y variables, so that the equation becomes x = f(y).
4. Solve for y.
5. Replace y with f-1(x).
6. Verify that f(f-1(x)) = x and f-1(f(x)) = x.

## 3. Is every function guaranteed to have an inverse function?

No, not every function has an inverse function. For a function to have an inverse, it must pass the horizontal line test, meaning that every horizontal line intersects the graph of the function at most once. If a function fails this test, it does not have an inverse function.

## 4. Can a function have more than one inverse?

No, a function can only have one inverse. This is because the inverse function must pass the vertical line test, meaning that every vertical line intersects the graph of the inverse function at most once. If a function has more than one inverse, it would fail this test.

## 5. What is the notation used for inverse functions?

The notation for inverse functions is f-1(x), where f is the original function. It is important to note that the "-1" in the notation does not indicate exponentiation, but rather denotes the inverse function.

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