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Video: Finding the Unknown That Makes an Exponential Function Increase on Its Domain

Alex Cutbill

What condition must there be on 𝑧 for 𝑓(𝑥) = (𝑧/7)^𝑥 to be an increasing function?

02:32

Video Transcript

What condition must there be on 𝑧 for 𝑓 of π‘₯ equals 𝑧 over seven to the power of π‘₯ to be an increasing function?

What’s the definition of an increasing function? A function 𝑓 of π‘₯ is said to be an increasing function if 𝑓 of π‘Ž is less than or equal to 𝑓 of 𝑏 whenever π‘Ž is less than 𝑏. This is the formal definition of an increasing function, but it turns out, like many things, that it’s easier to understand what it means for a function to be an increasing function by looking at its graph.

Let’s consider for a moment the related function 𝑔 of π‘₯ equals π‘Ž to the power of π‘₯. We can see that 𝑓 of π‘₯ it’s just a function with π‘Ž replaced by 𝑧 over seven. Let’s see if we can graph this function for different values of π‘Ž. When π‘Ž is equal to one, 𝑔 of π‘₯ is equal to one to the power of π‘₯ and one to the power of any number is just one. So we have the constant function 𝑔 of π‘₯ equals one, whose graph is shown.

When π‘Ž is greater than one, we have an exponential growth function; for example, 𝑔 of π‘₯ could be two to the power of π‘₯ or π‘Ž to the power of π‘₯. This function is increasing; as π‘₯ increases, the value of 𝑦 also increases. And looking at our formal definition, if we pick any two values of π‘Ž and 𝑏, where π‘Ž is less than 𝑏, then we see that 𝑓 of π‘Ž is less than or equal to 𝑓 of 𝑏. This shows that our formal definition is satisfied, but really, it’s best to look at the graph and see if 𝑦 increases as π‘₯ increases.

And finally, when π‘Ž is less than one, we have exponential decay. So this is a decreasing function: as π‘₯ increases, the value of 𝑦 decreases. To summarize, when π‘Ž is greater than one, the function is increasing, when π‘Ž is equal to one, the function is constant, and when π‘Ž is less than one, the function is decreasing.

We are interested in the condition that makes our function 𝑓 of π‘₯ an increasing function. We therefore need our value of π‘Ž to be greater than one. And looking at the definition of our function 𝑓 of π‘₯, we can see that our value of π‘Ž is 𝑧 over seven, so the condition we require is 𝑧 over seven is greater than one. And we can simplify this condition by multiplying both sides by seven to get the condition 𝑧 is greater than seven.