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(male narrator) In this video,
we will look at how we can find the inverse of a function.
If inverses do opposite things,
the regular function takes input and gives you an output.
The inverse takes output and gives you an input.
In other words, to find an inverse function,
what we'll want to do is switch...
the x and y...and then we will solve for y.
By the way, we need to remember
that f of x is really representing where the y is.
So if we had a function, like g of x here,
that g of x, we could think about as the y equals 5,
times the cube root of x, minus 6, plus 4.
And then, in order to start finding the inverse,
we need to switch any x and y values.
In other words, the y becomes an x; equals 5;
times the cube root of x, which becomes a y;
minus 6; plus 4.
In order to get to our inverse,
we simply have to solve this equation for y.
We know we must first isolate the radical
by subtracting 4 to get: x minus 4, equals 5,
times the cube root of y, minus 6.
And then divide both sides by 5,
x minus 4, over 5, is equal to the cube root of y, minus 6.
Now that the radical is alone,
we can clear a root with an exponent
by cubing both sides.
Let's leave this expression cubed,
as there is no advantage into multiplying it out.
We'll leave it as: x minus 4,
over 5 cubed, equals y, minus 6.
Finally, we can get the y alone
by adding 6 to both sides.
Now we end up with the quantity:
x minus 4, over 5 cubed, plus 6,
is equal to y.
This then is our inverse function.
We could say g inverse of x is equal to x minus 4,
over 5 cubed, plus 6.
Let's take a look at another example
where we're asked to find the inverse of a function.
Here, the h of x is representing
our y equals -3, over x, minus 1, minus 2.
We can then switch any x's and y's,
giving us x equals -3, over y, minus 1, minus 2,
and start solving for y.
We could start by isolating the fraction
and adding 2 to both sides...
giving us x plus 2, equals -3, over y, minus 1.
Now, to clear the fraction out, we can multiply
by that denominator-- y minus 1--on both sides.
Again, there's no advantage to multiplying this out,
so we'll keep it as y minus 1, times x, plus 2, equals -3.
Now that we've got the fraction taken care of,
we can start getting the y alone
by getting rid of the factor of x plus 2.
Dividing both sides by x plus 2...
leaves us with y minus 1,
equals -3, over x, plus 2.
To get the y completely alone,
all that's left is adding 1 to both sides.
And we get y equals -3, over x, plus 2, plus 1.
This then is our inverse function.
We can represent it as the inverse of h;
as h, -1, x; or h inverse of x; is equal to -3;
over x; plus 2; plus 1.
This is the inverse of the function h.
It will undo all the work that h does in the original function.
To find the inverse of a function,
we simply switch the x and y
and solve the resulting equation for y.