Find the inverse of the following function

Question

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

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,

Answer 1

$(y-4)^2=7-x$, 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.

Answer 2

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.