The inverse relation of the given function is [tex]f^{-1}(x)[/tex]=√(3x+5.25)+4.5.
The given function is f(x)=⅓ x²− 3x + 5.
We need to determine the inverse relation.
How do you find the inverse relation of a function?
In order to find the inverse of a quadratic equation, we first convert it into a perfect square, then we solve it further to bring it in the form of f(x)=a (x−h)²+k. We find the domain and range from here. After which, we represent our function f(x) as y and interchange the positions of x and y. We solve it further and thus get our required answer.
Now, y=⅓ x²− 3x + 5
Multiply both sides of an equation by 3.
That is, 3y=x²− 9x + 15
Simplify x²− 9x + 15 using the complete square method.
Now, (b/2)²=(9/2)²=20.25
Add and subtract 20.25 to x²− 9x + 15.
That is, x²− 9x+ 20.25-20.25+ 15
=(x-4.5)²-5.25
So, 3y=(x-4.5)²-5.25
Interchange x and y.
3x=(y-4.5)²-5.25
⇒3x+5.25=(y-4.5)²
⇒(y-4.5)=√(3x+5.25)
⇒y=√(3x+5.25)+4.5
Therefore, the inverse relation of the given function is [tex]f^{-1}(x)[/tex]=√(3x+5.25)+4.5.
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