Can someone explain both these questions, and elaborate on the process and what the counter examples mean? Textbook doesnt really explain it. Also, how common are they on exams (likey, not very?). Thanks.
A counter example is simply an example that shows a statement isn't true. For example, the "statement |x+y| = |x| + |y| for each real x and y". If we want to show that this is not true, we can find an example where the statement is false. We have |1 + (-1)| = |0| = 0, but |1| + |-1| = 1 + 1 = 2, so this shows that the statement is not true. For the statement "sin(x+y) = sin(x) + sin(y) for each real x and y". We have sin(π/2 + π/2) = sin(π) = 0, but sin(π/2) + sin(π/2) = 1 + 1 = 2.
Would someone give me a step by step process that works to solve, then graph the absolute value functions (Modulus)
My teacher has explained it poorly and i can understand some of the questions but get confused with some.
Also could i have an explanation of how to write the hybrid function of this graph.
Here's an example: [x^2+3x]
I'm assuming you mean |x
2 + 3x|. By the definition of the modulus function, |x
2 + 3x| = x
2 + 3x when x
2 + 3x ≥ 0, and |x
2 + 3x| = -(x
2 + 3x) when x
2 + 3x < 0. By sketching a graph of x
2 + 3x, you should see that x
2 + 3x ≥ 0 when x ≥ 0 and when x ≤ -3, and x
2 + 3x < 0 when -3 < x < 3. So you can write (using interval notation),
\\ <br />-(x^2+3x), & x\in(-3,0)<br />\end{matrix}\right.)
You should be able to sketch the two parts individually using your usual graphing techniques.
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