First of all try to figure out if the function is odd, even or neither.
To do that, check if f(-x) = f(x). If that's the case, then the positive part of the function is a mirror image of the negative part. If f(x) = - f(-x) then the positive part is identical to the negative part but reflected over the x axis. If none of the above is true, then the function behaves differently on either ends of the y-axis.
Secondly, you notice that sinx is a sinusoidal function, that varies between positive and negative. Therefore, as x increases above zero, you f(x) will take positive and negative values sinusoidally. However, you also notice that sin(x) is actually multiplied by e^x, which means that the peaks of the sine curve increase as x increases, so you get a wave that grows on the positive end of the graph.
Thirdly, you check the negative part, is it going to be similar? Well, returning to point 1, if you plug in f(-x), you get e^-x * sin(-x) = -e^-x*sin(x)
As you can see, the graph is neither even nor odd. So you expect it to behave differently on the negative end.
Why's that, you should ask. Well, if you look at the graph of e^x, as x becomes more negative, e^x falls in value, but always remains positive. So you fundamentally you still have a sine curve, but the peaks keep falling in height because you're multiplying by smaller numbers ( e^x) as you go the negative end of the x-axis.
Fourthly, you can try to figure out where the graph meets the x-axis, that's really simple, it's basically the values of x where sin(x) is zero.
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As for the rest of your questions, try what we just did. However, notice that if f(x) = x cosx, then f(-x) = (-x) cos(-x) = - x cosx = - f(x) so the function is odd, i.e. the negative bit is just a mirror image of the positive bit over the y- axis.
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