Mathematics: 4th round of free points cause i have nothing better to do 32432423432+234234=? 1-2323=?

4th round of free points cause i have nothing

Ответ:

drcarver

19.01.2021

Mathematics

If $h^{-1}(x)$ is the inverse of h(x) , then

$h\left(h^{-1}(x)\right) = x$

Given that h(x)=-(x-2)^2+7 , we have

$h\left(h^{-1}(x)\right) = -\left(h^{-1}(x)-2\right)^2 + 7 = x$

Solve for the inverse :

$-\left(h^{-1}(x)-2\right)^2 + 7 = x \\\\ -\left(h^{-1}(x)-2\right)^2 = x-7 \\\\ \left(h^{-1}(x)-2\right)^2 = 7-x \\\\ \sqrt{\left(h^{-1}(x)-2\right)^2} = \sqrt{7-x} \\\\ \left|h^{-1}(x)-2\right| = \sqrt{7-x}$

At this point, we have two possible solutions:

• If $h^{-1}(x)\ge2$ , then $\left|h^{-1}(x)-2\right| = h^{-1}(x)-2$ . Continuing with the equation, we have

$h^{-1}(x) - 2 = \sqrt{7-x} \\\\ h^{-1}(x) = 2 + \sqrt{7-x}$

• Otherwise, if $h^{-1}(x)$ , then $\left|h^{-1}(x)-2\right|=-\left(h^{-1}(x)\right)=2-h^{-1}(x)$ , in which case

$2-h^{-1}(x) = \sqrt{7-x} \\\\ h^{-1}(x) = 2 - \sqrt{7-x}$

Both solutions are simultaneously incompatible. For instance, when x = 0 we have $h^{-1}(0)=2+\sqrt7$ using the first solution, and $h^{-1}(0)=2-\sqrt7$ using the second one. Choosing one over the other depends on how you restrict the domain.

(A) If we want to stick with the first solution, we would required that x ≥ 2. In other words, we would have h(x) defined only when x ≥ 2; then $h\left(h^{-1}(x)\right)$ is defined only when $h^{-1}(x)\ge2$ , so that ...

(B) ... the inverse is

$\boxed{h^{-1}(x)=2+\sqrt{7-x}}$

• Domain: we have for all x that

$-(x-2)^2 \le 0$

$-(x-2)^2 + 7 \le 7$

which means the range of h(x) , and hence the domain of $y=h^{-1}(x)$ , is

$\left\{y \mid y \le 7\}$

• Range: we get this immediately from the domain restriction we chose earlier,

$\left\{x \mid x \ge 2\}$

See the attached plot - h(x) is shown as a dashed orange curve; h(x) with the restricted domain is shown in blue; the inverse $h^{-1}(x)$ corresponding to the restricted h(x) is shown in green; the other inverse is shown with a dashed red curve. You can see a geometrical properties of inverses: if you mirror the blue curve along the dotted line (y = x), you would get the green curve.

HEL

h(x)=-(x-2)^2+7
A) Restrict the domain of h(x) to make it invertible
B)Find h^(-1)(x), in term

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4th round of free points cause i have nothing better to do
32432423432+234234=?
1-2323=?

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