Explain how the graph of the function f(x) = (-1)/((x+1)^2) - 1 can be obtained from the graph o

Step 1: To obtain the graph of f(x)=(-1)/((x+1)^(2))-1 from the graph of y=(1)/(x^(2)), we can s

Explain how the graph of the function f(x) = (-1)/((x+1)^2)...

Explain how the graph of the function $f(x) = \frac{-1}{(x+1)^2} - 1$ can be obtained from the graph of $y = \frac{1}{x^2}$. Then graph f and give the (a) domain and (b) range. Determine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing.

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Transcript

00:01 All right, so in this problem, we have the function f of x equals 1 over x plus 1.

00:04 So that comes from the function 1 over x, and this plus 1 is not together with the x.

00:11 It's separate from that.

00:12 It's added afterward.

00:13 So that's going to be a transformation that moves up or down.

00:16 And when you move up or down, it kind of follows the logic of the number.

00:20 So plus 1 is going to go up 1 and minus 1 would go down 1.

00:24 So this transformation is going to go up 1 unit.

00:27 So that's affecting the horizontal asymptote, which normally would be at zero because the degree of the denominator is larger, but now it's shifted up 1 to y equals 1 like this.

00:41 The vertical asymptote will remain at 0 because 1 divided by 0 is undefined.

00:46 We're going to have our vertical asymptote at x equals 0 still.

00:51 That's going to remain the same.

00:52 That hasn't changed.

00:53 So the domain, the vertical asymptote affects the domain.

00:58 So the domain is everything except 0.

01:02 The range is affected by that horizontal asymptote.

01:05 So it's everything except 1.

01:09 And then as far as getting the graph, if you have a graphing calculator, you can use that or desmos, or you can just test different points...