Finding the image of $f: (0, \infty) \to \mathbb{R}, f(x) = \frac{4x}{x+1}$

Question

I need to find the image of the function $f: (0, \infty) \to \mathbb{R}, f(x) = \frac{4x}{x+1}$.

What I did was make the function in terms if y to end up with:

$x = \dfrac{y}{4-y}$

Now, I don't know how to go from here. I know that y can't equal $4$, but how do I prove that without using calculus.

Also, what is the double subset technique? Apparently I am supposed to use it for this question but I don't know exactly how to use it.

score 1 · Accepted Answer · answered Feb 06 '20 at 20:16

1

for $x\in (0,\infty)$, $0\lt\frac{x}{1+x}\lt1$. Thus $0\lt\frac{4x}{1+x}\lt4$

answered Feb 06 '20 at 20:16

Why didn't you multiply the denominator by 4? – bob657 Feb 06 '20 at 20:22
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I multiplied the whole inequality by 4. not numerator and denominator. – Prabath Hewasundara Feb 06 '20 at 20:24

1 Answers1