A function question about transformations...

Question

The function F(X)=(square root) X is transformed solely by using translations and reflections to produce g. The domain of g is all real numbers greater than or equal to -7, and the range is all real numbers less than or equal to  3. What is the equation that represents g. 

Answer 1

Alright, so this problem is purely a trial-and-error type or so it seems. However, you can actually make some analysis to make some educated guess.

 

There are 3 factors to consider when looking for a function of x using a square root: (I apologize for lots of words because as I said, this is an analytic problem)

 

1. Observing the simple f(x) = √x, it is also possible to have f(x) = -√x which is the mirror image (REFLECTION!) of the former. I'm just saying it's possible and more to that later.

 

2. We cannot have a negative value inside the square root (or at least in real Math), so in f(x) = √x, the domain is from 0 to positive infinity.

 

3. Since the first domain is 0, then it's safe to assume that the first range is also 0. (Proof: √0 = 0)

 

Considering all these things, let's check your given data.

 

First, the domain is x ≥ -7, meaning if the x is less than -7, the inside of the square root will turn into negative. (This is always the case when dealing with square root functions, excluding the ones with denominator as x)

 

Rearranging the domain, you may say that it's x + 7 ≥ 0, which is our goal for inside the square root, greater than or equal to 0. Meaning that is what's inside the square root.

 

√(x+7) *Proof: try some values greater than or equal to -7*

 

Next, the range. It says that it's from 3 down. (You: you said the first range should be 0. Yeah, if we're dealing with √x because it has no shift (TRANSLATION!), meaning this one has.)

 

Remember f(x) = y, so why not just assume first that y = 3. (Since the first range in a square root function is 0, refer to function 3, it won't hurt the assumption for now.)

 

Where is the square root thingy? Well try to put it there, maybe addition.

 

y = 3 + √(x+7) *Trial-and-error time!*

 

Test some values starting from -7, if you do (I hope), you will see that values of y is becoming higher than 3. Which is wrong because the range that we want is 3 DOWN.

 

So remember the Factor 1 above about mirror image or reflection whatever. -√x is possible. So does -√(x+7).

 

Making the function:

 

y = 3 - √(x+7) 

 

Check your conditions and hopefully it's right. (Just kidding, of course it is.)

 

*g(x) = 3 - √(x+7) whatever. lol.