If , where , which of the following must be true?
Answer Choices
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Wait, so f(x) = g(x-1) means I plug x-1 into g? So g(x-1) = |x-1| + 1. Since absolute value is always nonnegative, the smallest it can be is 0, so |x-1| + 1 ≥ 1. So f(x) is always positive. Answer C, right?
Yep that's how I did it too. Took me a second to see that f(x) is just g shifted, but your way is cleaner.
Is the trap here choice E? Because g(x) = |x| + 1 and f(x) = |x-1| + 1 are equal at x = 1/2 but not for all x. Just want to make sure I'm not missing something.
Yeah E is the trap for people who don't actually substitute. f and g aren't the same function, they're just related by a shift.
Got this one pretty quick, but only after reminding myself that |anything| is never negative. Once you see that, C is the only one that can be true for all x.
Can someone explain why D is wrong? I get that f(x) can equal 2 when x = 2 or x = 0, but not for every x. Is that the reason?
Right, "must be true" means for all values of x. f(1) = 1, so D fails.
What it tests
Your ability to evaluate and compose functions, including unusual defined operations.
Common trap
Applying function operations in the wrong order or ignoring domain restrictions.