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What condition must there be on 𝑧 for 𝑓(𝑥) = (𝑧/7)^𝑥 to be an increasing function?
What condition must there be on 𝑧 for 𝑓 of 𝑥 equals 𝑧 over seven to the power of 𝑥 to be an increasing function?
What’s the definition of an increasing function? A function 𝑓 of 𝑥 is said to be an increasing function if 𝑓 of 𝑎 is less than or equal to 𝑓 of 𝑏 whenever 𝑎 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 𝑥 equals 𝑎 to the power of 𝑥. We can see that 𝑓 of 𝑥 it’s just a function with 𝑎 replaced by 𝑧 over seven. Let’s see if we can graph this function for different values of 𝑎. When 𝑎 is equal to one, 𝑔 of 𝑥 is equal to one to the power of 𝑥 and one to the power of any number is just one. So we have the constant function 𝑔 of 𝑥 equals one, whose graph is shown.
When 𝑎 is greater than one, we have an exponential growth function; for example, 𝑔 of 𝑥 could be two to the power of 𝑥 or 𝑎 to the power of 𝑥. This function is increasing; as 𝑥 increases, the value of 𝑦 also increases. And looking at our formal definition, if we pick any two values of 𝑎 and 𝑏, where 𝑎 is less than 𝑏, then we see that 𝑓 of 𝑎 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 𝑥 increases.
And finally, when 𝑎 is less than one, we have exponential decay. So this is a decreasing function: as 𝑥 increases, the value of 𝑦 decreases. To summarize, when 𝑎 is greater than one, the function is increasing, when 𝑎 is equal to one, the function is constant, and when 𝑎 is less than one, the function is decreasing.
We are interested in the condition that makes our function 𝑓 of 𝑥 an increasing function. We therefore need our value of 𝑎 to be greater than one. And looking at the definition of our function 𝑓 of 𝑥, we can see that our value of 𝑎 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.
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