Determine the inverse function (Calculus 1 Homework)

Question

anonymous · Answer

determine the inverse function $$f ^{-1}$$ of f when $$f(x)=\sqrt{3+7x}$$ $$x \ge \frac{ -3 }{ 7 }$$

mimi_x3 · Answer

$$ y = \sqrt{3+7x}$$
$$ x = \sqrt{ 3+ 7y}$$
$$ \sqrt{3+7y} = x$$
$$ 3+ 7y = x^2$$
$$ 7y = x^2 - 3$$
$$ y = \frac{x^2-3}{7}$$

anonymous · Answer

@Mimi_x3 then it also says the answer is $$x \ge0$$

anonymous · Answer

why?

mimi_x3 · Answer

maybe cos of the condition?

zzr0ck3r · Answer

think of y = x^2
y = sqrt(x) is the inverse if we restrict x >= 0

zzr0ck3r · Answer

its only an inverse for x>=0 because we lose the negative if we make x negative
 
f(f^(-1)(x)) = x for all x is a must
and this is not the case unless we restrict the domain.
 this happens when we try and find the inverse of even roots

zzr0ck3r · Answer

the problem is here @Mimi_x3
$$\sqrt{3+7y}=x\\text{this implies that }x\ge0\\text{we loose that information in your next step}\3+7y = x^2$$

debbieg · Answer

Another way to think of the restriction on the domain of the inverse is to simply think about the range of the original function.

If you know what the range of $f(x)=\sqrt{3+7x}$ is (can you picture the graph?... this is just a transformation on $f(x)=\sqrt{x}$),   then you know that the domain of f(x) is $x\ge0$, so the range of the inverse must be $y\ge0$.

The inverse of a function has the DOMAIN of the original for its RANGE, and the RANGE OF of the original for its DOMAIN. That's what the inverse does, after all - it "flips" the funcion's domain and range values. :)

zzr0ck3r · Answer

correct:)

zzr0ck3r · Answer

so really asking for inverse without stating domain and co-domain is sort of pointless, we just assume that both are R

dan815 · Answer

i wanna be GREEN