Hi! Wanted to check logic here-
I am not sure of thr logic of the last option- I think domain=range, because:
- In domain, x can take any value
- In the range, f(x) can also be anything, i.e. 0, -ve, +ve, irrational number etc. becaue k can be anything.
You are right that the domain is all real numbers for the last option since we can plug in negatives, zero, and positives.
I understand your thought process, but since the function is only a function of x (only x varies), then k is a constant. We may not know what the constant is, but depending on the constant, the range is all numbers greater than or equal to k.
Suppose k = 1. Then, f(x) = |x| + 1. The smallest this function can be happens at x = 0 → f(0) = |0| + 1 = 0 + 1 = 1. Otherwise, we can get any value greater than that because |x| is greater than or equal to 0.
Suppose k = pi. Then, f(x) = |x| + pi. Similarly, the smallest this function can be is pi.
Even if we do not know what k is, its value is constant. It is impossible for f(x) to equal k-1 for example because f(x) = |x| + k = k-1 → |x| + k = k -1 → |x| = -1 → contradiction. Thus, we have found a value, namely k-1, that f(x) cannot attain. Thus, f cannot have all real numbers as its range.
I hope this helps!
Understood, thanks for your very detailed explanation. Esp. the contradiction example helped me understand this now, thanks Ryan!
Also just a follow on thought, this logic almost seems like the logic we use for finding min/ max for a quadratic using completing the square method. Basically we are saying that here, IxI can never be -ve, but can be 0/ positive, so minimum value has to be the constant
Yes, exactly! Happy to help
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