RANGE OF FUNCTIONS
A relation is a function if and only if there is exactly
one output for every input. A function has an independent
variable x , and a dependent variable y.
The range of a function refers to all of the possible values for the dependent
variable y that a function can have.
I. When a function is represented as a set of ordered pairs, the first
coordinate is the input and the second
coordinate is the output . The output values refer to the range.
Example: (1, 3) (4, 6) (5, 7) (7, 9) (9, 11)
In this case the range would be {3, 6, 7, 9, 11}
Range is denoted by f (x) when a function is given the name “f”.
II. When a function is represented or given as an equation, the range can be
found graphically or by using the
function’s inverse. (For more information about inverses refer to the Function
and Inverse Functions Handout.)
Finding the Range Graphically:
Depending on the type of function, the functions graph and range varies.
The range of functions is subject to change depending on whether the function
has vertical shifts or is reflected,
flipped, about the xaxis.
* The linear , cubic, logarithmic, and cube root functions are the exceptions. The
range for these functions is all
real numbers , written as (−∞,∞) or {y  y ε }.
*
The following functions have a set range and are in their standard state.
Linear Function Range: (− ∞,∞) or {y  y ε } 
Quadratic Function Range: [0,∞) or {y  y ≥ 0} 
Absolute Value Function Range: [0,∞) or {y  y ≥ 0} 

Square Root Function Range: [0,∞) or {y  y ≥ 0} 
Cubic Function Range: (− ∞,∞) or {y  y ε } 
Logarithmic Function Range: (− ∞,∞) or {y  y ε } 

Exponential Function Range: [0,∞) or {y  y ≥ 0} 
Cube Root Function Range: (− ∞,∞) or {y  y ε } 
The standard state function is represented as f (x) so when
a number is added or subtracted from f (x), y = f (x) ±
k where k is a number then a vertical shift occurs. For example, the standard
state function f (x) = x³ experiences
a vertical shift when f (x) = x³ + 5 or f (x) = x³  1.
Also, when a negative sign is placed before the standard state function, y = 
f (x), then the graph of the function
is reflected about the xaxis.
The following functions have changed by a vertical shift or were flipped about
the xaxis, and therefore
their range has been changed.
Here the standard state 
The standard state function, f (x) = x² has experienced a vertical shift. Therefore the range of the function changes to: Range: [− 2,∞) or {y  y ≥ 2} 
For rational functions the range depends on the function’s
horizontal asymptotes or slant asymptotes which
break up the range into two intervals .
Rational functions are composed of two polynomials being divided .
In this example, there is not a slant asymptote but there is a horizontal
asymptote.
Example: Horizontal asymptote: y = 0
Range: (− ∞,0),(0,∞) or {y  y ≠ 0} The asymptote excludes the value x=0 
Example: Horizontal asymptote: y = 2
Range: (− ∞,2),(2,∞) or {y  y ≠ 2} The asymptote excludes the value x=3 
The range can also be found by taking the inverse of the
function, written f^{ 1} (x), given and finding the domain
of the inverse.
To find the inverse of a function you will interchange the independent and
dependant variables and then solve
for the dependent variable.
Example:  y = x² + 6 
Interchange the x and y variables:  x = y² + 6 
Next, solve for y : 
The resulting equation is the inverse :
Find the domain of the inverse: Domain: (0.∞) or {y  y ≥
0}
Therefore the range of the original function, y = x² + 6, is: (0.∞) or {y  y ≥
0}
(If the function does not have an inverse, for example f (x) =  x  does not
have an inverse, use the other methods
listed above to find the range.)
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