Increase speed of vehicle - math question (javascript)

I have an endless runner game that I want to increase with speed as the game goes on. I want it to increase at a quicker rate at the beginning and a slower rate as time goes on. I figured a square root function would accomplish this but I realize it doesn’t. For example:

 speed = 10
 function gameSpeed(){
        	speed = Mathf.Sqrt(speed);
        	Debug.Log(speed);
        }
        
        InvokeRepeating("gameSpeed",0,1);

When I run this it prints: 3.16, 1.77, 1.33, etc. It does what I want except it gets smaller and smaller.

I also thought of a square function such as:

speed = 10
function gameSpeed(){
	speed = Mathf.Pow(speed, 2);
	Debug.Log(speed);
}

InvokeRepeating("gameSpeed",0,1);

And when I run this one it prints: 100, 1000, 100000000, etc. This one is at least increasing, except it is increasing at a slower rate at the beginning and a faster rate as more time goes on, opposite of what I want.

I just need the parent function of the equation, I know how to modify it to increase at the rates I want.

Why not just use a variable that increases by timeDelta then multiple that by a constant to get the rate of increase you require?

var speed = 10;
function gameSpeed(){
    speed += Time.deltaTime * 0.1;
}

I found my answer:

speed = 10
function gameSpeed(){
    speed += 100 / speed;
    Debug.Log(speed);
}
 
InvokeRepeating("gameSpeed",0,1);

In the “speed = Mathf.Sqrt(speed)” your speed variable is recursively defined in terms of its previous value, so speed becomes smaller EVERY second. Instead you need to take the square root of the time elapsed since the game started, as well as the starting speed. Here is a plot of what you have.

So instead you need to something like

     startSpeed = 10
     speed = 10
     time = 0				
     
     function gameSpeed(){
        speed = startSpeed + Mathf.Sqrt(time);      
        time++;         //Since this function is only called every second.
        Debug.Log(speed);
     }
     InvokeRepeating("gameSpeed",0,1);

Which will generate a speed plot that looks like