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Question 1120171: Suppose the point (−1,5) is on the graph of a function f(x).
(a) If f is even, then what other point (besides (−1,5)) must also lie on the graph of f(x)? Explain.
(b) If f is odd, then what other point (besides (−1,5)) must also lie on the graph of f(x)? Explain.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Formally, the function being even means f(-x) = f(x) for all x.
Informally, it means the function values are the same for inputs that are opposites.
In your example, the function contains the point (-1,5). That means f(-1)=5; f being even then means f(1)=5 also. So the point (1,5) is on the graph.
Formally, the function being odd means f(-x) = -f(x) for all x.
Informally, it means the function values are opposites for inputs that are opposites.
In your example, the function contains the point (-1,5). That means f(-1)=5; f being odd then means the function value will be the opposite for the opposite value of x -- i.e., f(1) = -5. So the point (1,-5) is on the graph.
Note that the graphs of even functions are symmetrical with respect to the y-axis, which means if (a,b) is on the graph then (-a,b) is also on the graph. This "mirror image" concept for even functions can be used to answer part a of your question: the mirror image with respect to the y-axis of (-1,5) is (1,5).
And graphs of odd functions are symmetrical with respect to the origin; that means if (a,b) is on the graph then (-a,-b) is also on the graph. And that concept gives you another way to find the answer to part b of your question.
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