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

What condition must there be on π§ for π(π₯) = (π§/7)^π₯ to be an increasing function?

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### Video Transcript

What condition must there be on π§ for π of π₯ equals π§ over seven to the power of π₯ to be an increasing function?

Whatβs the definition of an increasing function? A function π of π₯ is said to be an increasing function if π of π is less than or equal to π of π whenever π is less than π. This is the formal definition of an increasing function, but it turns out, like many things, that itβs easier to understand what it means for a function to be an increasing function by looking at its graph.

Letβs consider for a moment the related function π of π₯ equals π to the power of π₯. We can see that π of π₯ itβs just a function with π replaced by π§ over seven. Letβs see if we can graph this function for different values of π. When π is equal to one, π 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 π of π₯ equals one, whose graph is shown.

When π is greater than one, we have an exponential growth function; for example, π of π₯ could be two to the power of π₯ or π to the power of π₯. This function is increasing; as π₯ increases, the value of π¦ also increases. And looking at our formal definition, if we pick any two values of π and π, where π is less than π, then we see that π of π is less than or equal to π of π. This shows that our formal definition is satisfied, but really, itβs best to look at the graph and see if π¦ increases as π₯ increases.

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

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