Given the function : ( y=f(x)=frac5 x^2x^2+3 ) 1. Determine...

Given the function : ( y=f(x)=frac{5 x^{2}}{x^{2}+3} ) 1. Determine each of the following : (a) the ( x ) and ( y ) intercepts of the function (b) the type of symmetry, if any (c) the vertical and horizontal asymptotes, if any. 2. Compute the first derivative, and use your answer to determine where the function is increasing/decreasing; show your work on a number line.

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00:01 So we are given with function y equals to f of x which is equals to 5x square over x square plus 3 and we need to find x and y intercept with the function.

00:10 So x intercept will occur when y is 0.

00:13 So if you substitute y is 0, we'll have 5x square over x square plus 3.

00:20 And this will occur 0 when this numerator is 0 and therefore we get 5x square as 0 and that gives x as 0.

00:30 So this is our x intercept.

00:33 And now to find the y intercept, we have to put x as 0.

00:37 And again, if you substitute x as 0, we already know that is y 0.

00:42 And only intercept x and y axis it at 0 .0.

00:46 And since it is intersecting at 0 .0, it says that the curve touches at 0 .0.

00:53 So touches at 0 .0 at this point.

00:59 Okay.

00:59 Now the type of symmetry.

01:01 So the type of symmetry to find the symmetry about the y -axis, we can substitute x as minus -x or we can replace x with minus -x.

01:11 And if we replace it, we get 5 minus x -square will again be x -square, divided by minus x -square, again x -square plus 3.

01:20 And this says that this is equals to f -of -x.

01:23 And from this we can say the given function is symmetric about y -axis.

01:29 Since we get the same result, either we substitute minus x or we substitute x and we are getting the same y...

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