does the following Domain And Range statements are true

Question

 

I would like to know if following statement is ALWAYS TRUE

 

1.Does RANGE of root function ALWAYS ≥ 0 eg √x-9

 

2. Does RANGE logarithmic function ALWAYS equal TO ALL REAL NUMBERS eg lnx-3

 

3. Does DOMAIN of EXPONENTIAL function ALWAYS  equal TO ALL REAL NUMBERS eg X^2

Answer 1

To know the range, we evaluate the functions using the function's domain. 

 

 

1)

 

f(x) = √(x - 9)

 

The value under the square root must always be positive.  This means the domain of the function must be all real numbers greater than or equal to 9.

 

f(9) = √(9 - 9)

      = 0

 

f(11) = √(11 - 9)

        = 1.412

 

f(100) = √(100 - 9)

          = 9.5394

 

f(10000) = √(10000 - 9)

              = 99.9550

             

 

As you can see, at the starting domain, the f(x) value is 0.  As we increase the values in the domain, the f(x) values also increase to infinity. 

 

 

The following statement is true.

 

 

2)

 

g(x) = ln(x - 3)

 

The argument of a natural log must always be positive.   This means the domain of the function is all real numbers greater than 3.  So if we evaluate g(x), starting at x=3.000001 and increasing that x value to infinity,

 

g(3.000001) = ln(3.000001 - 3)

                   = -13.82

 

g(4) = ln(4 - 3)

       = 0

 

g(40) = ln(40 -3)

         = 3.611

 

g(400) = ln(400 - 3)

           = 5.9839

 

 

g(4000) = ln(4000 - 3)

             = 8.2933

 

g(40000000) = ln(40000000 - 3)

                    = 17.5043

 

The range is NOT ALWAYS all real numbers greater than or equal zero, because when x=3.0000001, the value of g(x) is negative.

 

 

3)

 

r(x) = x²

 

This function is a parabola.  In a parabola, the domain is all real numbers.

 

In an exponential function such as r'(x) = 2^x, the domain is also real numbers.