Question and Concept Help

[MrBrillsbury]

Hey!

Can somebody help me solve the following question?

The graph of f(x)=5+4x-x^2 has its domain restricted to x∈(-∞, a], where a is the largest possible value such that the inverse function exists. Determine the value of a.

In addition, if someone can explain how to find a based on any equation, with any given domain (whether I need to find x or y), and any such condition (largest possible value such that...), that'd be much appreciated!

Thank you!

[Quantum44]

Hey!

Can somebody help me solve the following question?

The graph of f(x)=5+4x-x^2 has its domain restricted to x∈(-∞, a], where a is the largest possible value such that the inverse function exists. Determine the value of a.

In addition, if someone can explain how to find a based on any equation, with any given domain (whether I need to find x or y), and any such condition (largest possible value such that...), that'd be much appreciated!

Thank you!

The inverse function exists only when f(x) is a one-to-one function so you need to restrict the domain to the turning point. Thus a will be the x-value of the turning point of f(x).

[Shadowxo]

In addition to what Quantum said,
For this question, you can use two ways.  Since you know the general shape and it's a parabola, you know it's 1:1 from negative infinity to the turning point (or the turning point to infinity, but not both). So you can put it into turning point form and use the x value from there.
Alternatively, you can use the derivative - better for non-parabolas. It's 1:1 between negative infinity and the lowest x value where there is a max/min and between two max/mins etc.
eg: imagine a cubic that has two turning points, a max and a min, at x=-1 and x=2. From x=negative infinity to x=-1,it will be one to one, and between x=-1 and x=2, it'll be one to one, and between x=2 and x= infinity, it'll be 1:1.
So dy/dx=0 where x=2, so that's the turning point, and it's 1:1 between x=-infinity and x=2, therefore a=2