Tommy B. answered • 09/12/16
Tutor
5.0
(212)
Physics and Mathematics Tutor
One way to think about inverse functions is how they operate. It isn't always possible to algebraically write down an inverse function. Sometimes it is just messy. Sometimes it is impossible.
Remember that functions take values from one set of numbers and map them to another set of numbers. It's like a machine that takes x-values and turns them into y values. We write the output of that machine as y=f(x). For example, the function f(x) = 2x doubles the value of any x you put inside. That means f(1)=2, f(3)=6, f(10)=20, etc. Functions are rules that tell us what to do to the input value.
What if we want to go the other way? What if we have the output of a function but we don't know the input value x that gave us the output? For instance, my function f(x)=2x. Suppose I told you f(x)=4. You would say f(x)=2x, so 2x=4 and you would solve the equation finding x=2. Check it: f(2)=2*2=4.
Looking for an x value that gives the output value f(x) is called inverting. It is convenient to introduce a letter to hold the output. For instance, f(x)=y. We say the inverse function is the one that takes y-values and gives back x-values. We write that as f-1(y)=x. For my example f(2)=4 we would say f-1(4)=2. It goes just backwards.
In your problem, there is an unknown k in the function f(x) = kx3 - 1. We're told that f-1(15)=2. In other words, y=15 and x=2. The input of f is x=2 and the output of f is 15. So, we have the input and the output. Only the value of k is unknown. Since f-1(15)=2, we know f(2)=15. To find k, we can just plug x=2 into f(x)! That means
f(2) = k * (2)3 - 1 = 15.
Simplifying a bit gives
8k - 1 = 15.
Now we can solve for k in the following manner:
8k - 1 + 1 = 15 + 1
8k = 16
k = 2
I hope this helps you understand how inverse functions work. Good luck!
Remember that functions take values from one set of numbers and map them to another set of numbers. It's like a machine that takes x-values and turns them into y values. We write the output of that machine as y=f(x). For example, the function f(x) = 2x doubles the value of any x you put inside. That means f(1)=2, f(3)=6, f(10)=20, etc. Functions are rules that tell us what to do to the input value.
What if we want to go the other way? What if we have the output of a function but we don't know the input value x that gave us the output? For instance, my function f(x)=2x. Suppose I told you f(x)=4. You would say f(x)=2x, so 2x=4 and you would solve the equation finding x=2. Check it: f(2)=2*2=4.
Looking for an x value that gives the output value f(x) is called inverting. It is convenient to introduce a letter to hold the output. For instance, f(x)=y. We say the inverse function is the one that takes y-values and gives back x-values. We write that as f-1(y)=x. For my example f(2)=4 we would say f-1(4)=2. It goes just backwards.
In your problem, there is an unknown k in the function f(x) = kx3 - 1. We're told that f-1(15)=2. In other words, y=15 and x=2. The input of f is x=2 and the output of f is 15. So, we have the input and the output. Only the value of k is unknown. Since f-1(15)=2, we know f(2)=15. To find k, we can just plug x=2 into f(x)! That means
f(2) = k * (2)3 - 1 = 15.
Simplifying a bit gives
8k - 1 = 15.
Now we can solve for k in the following manner:
8k - 1 + 1 = 15 + 1
8k = 16
k = 2
I hope this helps you understand how inverse functions work. Good luck!
