Question

find the range of x^2÷1+x^2 full step
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Answer 1

Answer:

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Answer 2

Step-by-step explanation:

You can compute the expression for any (real) value of xx.

For the codomain, try and solve the equation

y=x21+x2y=x21+x2

This becomes y+x2y=x2y+x2y=x2, so

x2=y1−yx2=y1−y

Since x2≥0x2≥0 and every nonnegative number has a square root, the solution will be found if and only if

y1−y≥0y1−y≥0

which amounts to 0≤y<10≤y<1. So the range is the interval [0,1)[0,1).

With calculus it’s easier: we see that, with f(x)=x2/(1+x2)f(x)=x2/(1+x2),

limx→−∞f(x)=limx→∞f(x)=1limx→−∞f(x)=limx→∞f(x)=1

and also

f′(x)=2x(1+x2)2f′(x)=2x(1+x2)2

so that ff is strictly decreasing over (−∞,0](−∞,0]and strictly increasing over [