SOLUTION: Determine whether the given function is one-to-one. If it is one-to-one, find its inverse.
f(x) = 7x^2 + 3, x ≥ 0
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Question 316861: Determine whether the given function is one-to-one. If it is one-to-one, find its inverse.
f(x) = 7x^2 + 3, x ≥ 0
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Simplest way to figure this out is to graph it.
The graph would look like this:
You can see that, for all values of x, not just positive values of x, there is only one value of y for each value of x.
This makes the equation a function.
In this graph, y = f(x) = x^2 + 3
To find the inverse of the function, you would replace y with x and x with y and then solve for y.
Your equation is y = x^2 + 3
replace y with x and x with y to get:
x = y^2 + 3
solve for y to get:
y = +/- sqrt(x-3)
A graph of the inverse equation would look like this:
You know this is an inverse equation, because it reverses the operation.
The inverse equation only exists when x >= 0.
When x = 5, the original equation gets you 5^2 + 3 = 25 + 3 = 28
In the inverse equation, when x = 28, the inverse equation gets you sqrt(28-3) = sqrt(25) = 5.
With the original equation you go from 5 to 28.
With the inverse equation you go from 28 to 5.
While your original equation is a function, your inverse equation is not.
In a function, every value of x must yield only 1 value of y.
If any 1 value of x yields more than 1 value of y, then the equation is not a function.
In that case, it is called a relation.
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