Question

Can someone please help me with number 40. Plss

Answer 1

`f(x)=√(x+2)-2`

Explanation:

So for the graph as shown in your image, it either appears to be logarithmic or a radical function.

Considering how it doesn't have a long tail at the end going towards negative infinity, it's most likely a radical. I attached an image of a square root, and a logarithmic function to really demonstrate the difference (notice the long "tail" at the end of the logarithmic function)

Generally whenever you have a square root, you have it in the form:
`f(x)=a\sqrt[n]{x-h}+k`

where (h, k) will be the minimum point (only if degree is even, since odd degrees are defined for all values of x, so there is no "minimum" value)

In your graph you provided, it appears that the minimum point is (-2, -2), so h=-2, and k=-2

Plugging this into the equation, you get:
`f(x)=a√(x-(-2))-2=a√(x+2)-2`

Now to make sure there isn't some value in front of the radical we can calculate some values (besides the minimum, sqrt(2-2) = 0, so any value of a will result in a * 0 - 2 = 0 - 2 = -2, thus all values of a work specifically at the minimum, so we can't use that point)

So by looking at the graph we see the point: (-1, -1)

Using the equation, plug in the values to get:
`-1=a√(-1+2)-2\\-1=a√(1)-2\\-1=a-2\\1=a`

So we know a is just 1, thus we don't have to explicitly write it in our equation:
`f(x)=√(x+2)-2`