Understanding Inverse Functions: How to Find f^-1(y)

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In summary, the inverse function of a function is the function that undoes what the original function does.
  • #1
sonya
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if ur given a function f(x) and ur asked to find f^-1(y)...r u supposed 2 solve ur original eqn for y and then take the inverse of that? or isn't that just the same thing neways?...
 
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  • #2
What language precisely is that?

Anyway-

I don't know exactly how you have been taught to find inverse functions- there are several ways to arrive at the same result.

The way I like is this: Swap x and y.

Yes, that's it: If f(x)= y then f-1(y)= x.

If f(10)= 0 then f-1(0)= 10.

If f(x) is given by y= f(x)= 3x- 4 then the inverse function is given by x= 3y- 4.

Oh, there is one tiny other thing you might want to do:
Since we prefer to write f(x)= ... or f-1(x)= ...,
you might want to solve for y!

Since x= 3y- 4, 3y= x+ 4 and y= f-1(x)= (x+4)/3.

Notice the key point: what f(x) "does", f-1(x) "undoes".
Where f(x) is "multiply by 3 then add 4", f-1(x) says "subtract 4, then divide by 3". Each step is reversed ("add 4" instead of "subtract 4" and "divide by 3" instead of "multiply by 3") and the order is also reversed. Of course: when I go to work in the morning, I put on my shoes, then go out the door, then lock the door behind me. When I come home in the evening, I first UNlock the door, then go in through the door, then take off my shoes. Each operation is reversed and the order is reversed.

Of course, you can't always "solve" for the inverse function.

If f(x)= ex then f-1(x)= ln(x) because that is the way ln is DEFINED- as the inverse function to ex.
 
  • #3
Originally posted by HallsofIvy

Since x= 3y- 4, 3y= x+ 4 and y= f-1(x)= (x+4)/3.

ok..i think i understand now but a quick question
where x = 3y -4 that can also be called f-1(y) rite?
 
  • #4
o..never mind that question...i get it now
thx a lot!
 

1. What is the purpose of a function in mathematics?

A function in mathematics is a rule or relationship that assigns one input value to one output value. It is used to describe the relationship between two or more quantities and to model real-world situations.

2. How do you determine the domain and range of a function?

The domain of a function is the set of all possible input values for which the function is defined. It can be determined by looking at the restrictions on the input values in the function's equation or graph. The range of a function is the set of all possible output values for the corresponding input values. It can be determined by looking at the vertical values on the function's graph or by solving for the output values in the function's equation.

3. What is the difference between a linear function and a nonlinear function?

A linear function is a function where the output varies directly with the input. In other words, the output changes by a constant rate for every unit change in the input. A nonlinear function, on the other hand, does not follow this pattern and the output changes at a varying rate for different input values.

4. How do you graph a function?

To graph a function, you first need to determine the domain and range of the function. Then, choose a few input values and use the function to calculate the corresponding output values. Plot these points on a coordinate plane and connect them with a smooth curve. Additionally, you can use transformations, such as shifting, stretching, or reflecting the graph to better understand the behavior of the function.

5. How is a function related to its inverse function?

A function and its inverse function are two functions that "undo" each other. In other words, if the input of one function is the output of the other, and vice versa, then they are inverse functions. The inverse function can be obtained by switching the input and output variables in the original function's equation. The domain and range of the inverse function are also switched from the original function.

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