Question Video: Finding the Domain and Range of Exponential Functions Mathematics

Video Transcript

Find the domain and the range of the function 𝑓 of 𝑧 is equal to one over eight raised to the power of the absolute value of 𝑧.

To find the domain and range of the given function, the first thing we notice that our function is an exponential function of the form 𝑦 is equal to 𝑏 raised to the power 𝑥, where the base 𝑏 is less than one. And this represents exponential decay. In our case, 𝑏 is equal to one over eight. That’s our base. Now, for an exponential decay function of this form shown, that’s 𝑦 is equal to 𝑏 raised to the power 𝑥, there are no values of 𝑥 for which the function is undefined. And the same goes for our function 𝑓 of 𝑧 is one over eight raised to the power of the absolute value of 𝑧. There are no values of 𝑧 for which this function is undefined. Hence, the domain of our function 𝑓 of 𝑧 is the set of real numbers ℝ.

Now the fact that our exponent is the absolute value of 𝑧 simply means that our exponent is always positive and the function itself is therefore always positive. And we can see this quite clearly on a sketch of our function. We have a positive base to a positive exponent. But this is actually only half of the story. The exponent in the general exponential decay function can take both positive and negative values, whereas in our case, we have only positive-valued exponents. And for 𝑧 greater than zero, our function is in fact decreasing. As 𝑧 gets larger and larger, 𝑓 of 𝑧, that’s one over eight raised to the power of the absolute value of 𝑧, gets smaller and smaller. In fact it approaches zero but never reaches zero.

When 𝑧 is equal to zero, our function takes the value of one. That is, 𝑓 of zero is one over eight, raised to the power of the absolute value of zero. And anything raised to the power of zero is equal to one. Hence, our function passes through the point zero, one. Now to the left of 𝑧 is equal to zero, as 𝑧 becomes large and negative, our exponent, the absolute value of 𝑧, again becomes very large and positive. And so noting we can rewrite our function 𝑓 of 𝑧 is one over eight raised to the absolute value of 𝑧, the denominator becomes very, very large as 𝑧 becomes very large in either the positive or negative directions. And so for large values of 𝑧 in either direction, our function tends to zero.

And so we see that the maximum value our function can take is one. The function is never below the 𝑧-axis. So 𝑦 never takes negative values. And hence, the range of the function 𝑓 of 𝑧 is the interval that’s open from zero and closed at positive one. And hence, for the function 𝑓 of 𝑧 is equal to one over eight raised to the power the absolute value of 𝑧, the domain of 𝑓 of 𝑧 is the set of real numbers ℝ. And that’s the set of possible input values. And the range of 𝑓 of 𝑧 is the half open interval between zero and one, where it’s open at zero and closed at one. So 𝑓 of 𝑧 equals zero is not included in the range, and 𝑓 of 𝑧 equal to one is included. Remember the range of a function is the set of all possible output values.