On the Prepswift Tickbox Quiz #5, I came across a couple of function problems that confused me. In the first question (#22), I assumed that if two functions are equal and a specific output is given, then both functions should produce the same output. That logic worked, and I got the question right. However, the second problem (#23) had a similar format, so I applied the same reasoning, but it turned out to be incorrect. What is the difference between these two function questions, and why doesn’t the rule I used in the first problem apply to the second one?
This assumption is wrong. Just because f(x) = f(x–1) leads to the same output for all x in that specific function, it doesn’t mean every function that looks like f(x) = f(some expression) will always behave the same.
In the first question, f(x) = f(x–1) and f(5) = 4 means the function keeps giving the same value as you go back:
f(5) = f(4) = f(3) = … — so yes, it ends up being constant (always 4).
But that only works because of the specific rule f(x) = f(x–1).
Other function rules like f(x) = f(–x) or f(x) = f(x+1) don’t necessarily force the function to be constant. For example:
- If f(x) = f(–x), then f(2) = f(–2), but that doesn’t tell you anything about f(3) or f(1).
- So just having a format like f(x) = f(something) doesn’t automatically mean the function is constant.
You can’t assume a function is constant just because it’s equal to itself in a certain format. You have to look at how the equation connects the values.
Solution for the second question
The yellow arrow mark is supposed to indicate that I applied the rule. The black arrow mark indicates that I used the values to back-solve. The equations inside the teal box are the solutions.
Hi, could you explain how you got here. I am confused how you determined the relationship between the functions with only f(1) = 2. where did the four squared come from in f(9)? is it because (f(3)^2 is actually (3+1)^2? I’m very confused.
First: what do we know?
f(1) = 2 → the output when x = 1 is 2
f(3x) = (f(x))^2 → the output when 3x is input is the same as the square of the output when x is input.
Let’s take a step back. Instead of f(9), can you use the two rules above to find f(3) instead? Maybe we can substitute something in the second equation?
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