The function f(x) = 7x^2 − 4x + 9 is not invertible, however, the domain can be restricted so that the function is invertible only on the restricted domain.

Question

There are many ways to do this. One way is to only use the set of positive, real numbers of the form [a,∞). Determine a value of a so that it is the smallest possible, positive real number where the function is invertible if a ≤ x < ∞. Using your value of a determine the domain for which f(x) is invertible. Provide your answer in interval notation.

 

Any help would be greatly appreciated!

Answer 1

If you think about the horizontal line test, the reason f(x) is not invertible is because its graph is a parabola: so if you draw a horizontal line you can hit the graph twice. You can fix this by cutting the parabola in half, and only taking the right half. In other words, find the x-value of the vertex, and from there to the right will be your domain.

 

The vertex of a parabola is always at

 

x = -b/(2a)

 

For this graph, that evaluates to x = 4/14, or 2/7. So your restricted domain is [2/7, ∞).