### 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?

Whatβs 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 itβs easier to understand what it means for a function to be an increasing function by looking at its graph.

Letβs consider for a moment the related function π of π₯ equals π to the power of π₯. We can see that π of π₯ itβs just a function with π replaced by π§ over seven. Letβs 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, itβs 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.