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Expert: - 10/12/2008


Question

Questions on Function
Hi, Please see the image attached. How do you work out the range and domain of functions in general? Thank you

Answer

Domain and Range
Questioner:   Sally
Category:  Advanced Math
 
Subject:  Functions



Hi, Please see the image attached. How do you work out the range and domain of functions in general? Thank you  
............................
Hi, Sally,

Well, if you ask 'in general', try something like this:

If you have a FRACTION, exclude any values of x that make a denominator zero.  These would usually be SPECIFIC values.  Write  x /= <whatever number(s) make the bottom zero>

If you have a square root, exclude any value(s) that make the radicand negative.  This would usually be some interval(s).   Write  x >= (some number) or  x <= (some number)

The RANGE of a function is the domain of its inverse.  Sometimes this is tricky.  But you can do this for your   g(x) = 3x^2.

Find the inverse.  Here is the standard procedure:

Write  y = 3x^2.

Switch  x and y:

X = 3y^2.

Solve for y:

Y = sqrt(x/3)

Now the domain is  x >= 0, so the range of g was  y >= 0.  {Switching back, of course.}

For your example 6.

(i). Domain of a polynomial is all x.  But for the range, you have some work to do.  This is a quadratic -- its graph is a parabola with vertex at the bottom.  Find the vertex.  Whatever y-coordinate you get;  write   y >= (that number) as the range.

(ii) You must have  x^2 - 3x + 2  >= 0.  Do you remember solving inequalities?  That's what you do here.  

(iii) Find the vertex again for  y = x^2 - 3x + 2.  If it is below zero, then your bottom value is zero.  (Remember what absolute value means.)

Use all this to sketch.  I have attached a picture.

Paul Klarreich

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