# Find the inverse of the following function

$y = 4 + \sqrt{7 - x}$

for my answer I got $-x^2 +8x -9 = y$, $y$ less than or equal to 7

the actual answer is $y = 7 - (x-4)^2$, $x$ greater than or equal to 4

my answer is correct except the domain isn`t. I thought that for inverse functions the domain and the range switch. There fore since the domain in the original function is $x$ less than or equal to 7 this would become the range in the inverse.

Thanks,

$$(y-4)^2=7-x$$, which gives $$x=7-(y-4)^2$$ and since in the given $$y\geq4$$,

we get the answer: $$y=7-(x-4)^2$$, where $$x\geq4$$.

Definition: $$f$$ and $$g$$ defined inverse functions if:

1. $$D(f)=R(g)$$;

2. $$D(g)=R(f)$$;

3. $$f(g(x))=x$$ for all $$x\in D(g)$$ and $$g(f(x))=x$$ for all $$x\in D(f)$$.

In our case $$f(x)=4+\sqrt{7-x}$$, where $$x\leq7$$ and $$g(x)=7-(4-x)^2$$, where $$x\geq4$$.

$$D(f)=(-\infty,7]=R(g)$$, $$D(g)=[4,+\infty)=R(f)$$.

Also $$f(g(x))=x$$ for all $$x\geq4$$ and $$g(f(x)=x$$ for all $$x\leq7$$,

which says that $$f$$ and $$g$$ are inverse function by definition.

You have correctly identified the range of the inverse function. But the definition of a function requires a domain, which you did not specify. The range is something you can derive from the definition, not (usually) part of the definition itself.