Domain and Range of Exponential and Logarithmic Functions
Recall that thedomainof a function is the set of input or-values for which the function is defined, while therangeis the set of all the output or-values that the function takes.
A simple exponential function likehas as its domain the whole real line. But its range is only thepositivereal numbers,never takes a negative value. Furthermore, it never actually reaches, though it approaches asymptotically asgoes to.
If we replacewithto get the equation, the graph gets reflected around the-axis, but the domain and range do not change:
If we put a negative sign in frontto get the equation, the graph gets reflected around the-axis. We still have the whole real line as our domain, but the range is now the negative numbers,.
Now, consider the function. When,must be a complex number, so things get tricky. For this lesson we will require that our bases be positive for the moment, so that we can stay in the real-valued world.
In general, the graph of the basic exponential functiondrops fromtowhenasvaries fromtoand rises fromtowhen.
The exponential function, can be shiftedunits vertically andunits horizontally with the equation. Then the domain of the function remains unchanged and the range becomes.
Example 1:
Find the domain and range of the function.
Graph the function on a coordinate plane.
The graph is nothing but the graphtranslatedunits to the left.
The function is defined for all real numbers. So, the domain of the function is set of real numbers.
Astends to, the value of the function also tends toand astends to, the function approaches the-axis but never touches it.
Therefore, the range of the function is set of real positive numbers or.
Example 2:
Find the domain and range of the function.
Graph the function on a coordinate plane.
The graph is nothing but the graphcompressed by a factor of.
The function is defined for all real numbers. So, the domain of the function is set of real numbers.
Astends to, the value of the function tends to zero and the graph approaches-axis but never touches it.
Astends to, the function also tends to.
Therefore, the range of the function is set of real positive numbers or.
The inverse of an exponential function is a logarithmic function.
A simple对数函数 whereis equivalent to the function. That is,is the inverse of the function.
The functionhas the domain of set of positive real numbers and the range of set of real numbers.
Remember that since the logarithmic function is the inverse of the exponential function, the domain of logarithmic function is the range of exponential function, and vice versa.
In general, the functionwhereandis a continuous and one-to-one function. Note that the logarithmic functionisnot definedfor negative numbers or for zero. The graph of the function approaches the-axis astends to, but never touches it.
Therefore, the domain of the logarithmic functionis the set of positive real numbers and the range is the set of real numbers.
The function rises fromtoasincreases ifand falls fromtoasincreases if.
The logarithmic function,, can be shiftedunits vertically andunits horizontally with the equation. Then the domain of the function becomes. However, the range remains the same.
Example 3:
Find the domain and range of the function.
Graph the function on a coordinate plane.Remember that when no base is shown, the base is understood to be.
The graph is nothing but the graphtranslatedunits down.
只有积极的真正的num的函数定义bers. So, the domain of the function is set of positive real numbers or.
The function takes all the real values fromto.
Therefore, the range of the function is set of real numbers.
Example 4:
Find the domain and range of the function.
Graph the function on a coordinate plane.
The graph is nothing but the graphtranslatedunits to the right andunits up.
Astends to, the function approaches the linebut never touches it. Astends tothe value of the function also tends to. That is, the function is defined for real numbers greater than. So, the domain of the function is set of positive real numbers or.
The function takes all the real values fromto.
Therefore, the range of the function is set of real numbers.