Range and Domain of Functions

The graph of a function is a big help in finding the range and domain of a function. The Table feature is also useful.

Find the range and domain of the polynomial funtion f(x) = x³ + 2x² - 3x - 6

• Graph the function (see Graphing Functions)
  
  

• Since the graph has only local maximum and minimum values we know the range is all real numbers (the function values (y-coordinates) are all the real numbers.)
  
  
• The function also has a value for every real number we use for x. Press the TABLE key and then choose the TABLE option. Scroll up and down the table, using the up or down arrow keys, and observe that every x-value in the table has a corresponding value for y. From this we know that the domain of the function is also all real numbers.
  

---

Find the range and domain of the absolute value funtion f(x) = |2x - 5|

• Graph this function;
  
  

• We can see from the graph that this function has a minimum value, but does not have a maximum value. The range then will be all real numbers greater than or equal to this minimum number.
  
  
• We find the minimum value is 0 (see Minimum Values of Functions). The range is then all nonnegative real numbers.
  
  
• The function has a value for every real number we use for x. Press the TABLE key and then choose the TABLE option. Scroll up and down the table, using the up or down arrow keys, and observe that every x-value in the table has a corresponding value for y. From this we know that the domain of the function is also all real numbers.
  

---

Find the range and domain of the rational funtion f(x) =

• Graph the function;
  
  

• This graph has a minimum and also seems to have a horizontal asymptote. The range of y values then will be all numbers greater than or equal to the minimum value and less than the location of the horizontal asymptote. Again if you look at the TABLE you can see the range of y-values that this funtion has.
  
  
• We find that the minimum value of the function is -9. We also locate a horizontal asymptote at y=2 (see Asymptotes of Rational Functions).
  
  
• We can also see from the graph that the function has a value for every real number we use for x. This can sometimes be misleading. Since the function definition has a denominator we should be concerned that there is a value of x that makes the denominator zero. If so this number would not be in the domain. Since the denominator 2x² + 1 is always greater than zero we know this does not happen. To double check this scroll through its TABLE and observe that every x value in the table has a corresponding value for y. The range is all real numbers. Look at the y-values to see if they agree with what we found above for the range.
  

---

Find the domain and range for the radical funtion f(x) =

• Graph the function
  
  

• We can see that the graph seems to have no maximum value, but has a minimum value of 0 (do this by finding the minimum value or looking through the function table). Therefor the range of the funtion is all nonnegative real numbers.
  
  
• The graph also shows that the function does not have values around 0. We can use TRACE or TABLE or the location of the local minimum values to conclude that the range is all real numbers less than or equal to -4 or greater than or equal to 5. Note that the function table has ERROR for the value of y1 corresponding to numbers strictly between -4 and 5. These are values that make x² - x - 20 negative, which do not have square roots.