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Finding an inverse function

  1. Sep 17, 2015 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    See below

    3. 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?
     
  2. jcsd
  3. Sep 17, 2015 #2

    andrewkirk

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    Gold Member

    Just make u the subject of the equation.
     
  4. Sep 17, 2015 #3

    Ray Vickson

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    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!
     
  5. Sep 18, 2015 #4
    So the inverse would be u=8+0.85*x?
     
  6. Sep 18, 2015 #5

    Mark44

    Staff: Mentor

    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.
     
  7. Sep 18, 2015 #6

    Mark44

    Staff: Mentor

    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.
     
  8. Sep 18, 2015 #7

    Ray Vickson

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    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?
     
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