Range and Domain of Exponential Function

In the previous post about inverse function, we learned to find the inverse function of an exponential function. Now it is time to discuss how to find the range and domain of exponential function. Standard exponential function can be expressed as y=a^u+c where u is a function of y and a>1 and 0<a<1.


Sample forms:

y=3^x+4 y=2^{x-2} y=4^{x-1}+3

The graph of a standard exponential equation is shown above

Graphically, we can say that the domain is the set of all real numbers since the graph extends from positive to negative x-axis. The range varies in the base and the constant of the function.


Worked Problem 1:

Find the range of y=3^x+4


We know that the domain is the set of all real numbers as mentioned above.

For the range:

Let x=0, then y=5

Let x=1, then y=6

Let x=-1, then y=4 1/3

Let x=-2, then y=4 1/9

As we lower the value of x, the value of y will almost approach to 4 but will not actually the line y=4.

From here we can say that the range of this function is {y│y>4} or in symbol  (4,+∞).


Worked Problem 2:

Find the range of y=3^{x-1}+1


Let x=1, then y=2

Let x=0, then y=4/3

Let x=-1, then y=10/9

Going further from here we can see that the range of this function is {y│y>1} or in symbol  (1,+∞) and the domain is the set of all real numbers.


Solution by logarithm:

In logarithmic function y=\log u this function is real if u>0. We know that the base of this function is 10.

Important Rule:

To get the domain solve for y.

To get the range solve for x.


Following these two rules, we will be able to identify the range without trial and error.

From problem 1,

Find the range of y=3^x+4

Solution by logarithm:

Solve for x since we are looking for the range:

y=3^x+4 3^x=y-4 \log 3^x=\log (y-4) x\log 3=\log (y-4) x=\displaystyle\frac{\log (y-4)}{\log 3}

As mentioned above y-4>0 making it {y│y>4} as the range of this equation.

If you have a keen observation you might have notice this. Given an exponential function y=a^u+c where u is linear in nature, the range is {y│y>c} and the domain is always the set of all real numbers.


Worked Problem 3:

Find the range of y=2^{2x}-1

This time we don’t need to solve the range by assigning the values to x or using the logarithm method. All we need to do to look closely that 2x is linear. Close our eyes and Voila! The range is {y│y>-1} or (-1,+∞)


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