Question

write the inverse of $d(x)=(x + 8)^2$? inverse of $d(x)=pm$ square root $\square+\square$

Explanation:

Step1: Let \( y = d(x)=(x + 8)^2 \)

To find the inverse, we first swap \( x \) and \( y \), so we get \( x=(y + 8)^2 \).

Step2: Solve for \( y \)

Take the square root of both sides: \( \pm\sqrt{x}=y + 8 \). Then, isolate \( y \) by subtracting 8? Wait, no, wait. Wait, the form given is "±square root [ ] + [ ]". Wait, let's re - express. From \( x=(y + 8)^2 \), we can also think of solving for \( y \) as \( y=\pm\sqrt{x}-8 \)? Wait, no, wait. Wait, the given structure is "Inverse of \( d(x)=\pm \) square root [ ] + [ ]". Wait, maybe I made a mistake. Let's start over.

Let \( y=(x + 8)^2 \). To find the inverse function, we solve for \( x \) in terms of \( y \) and then swap \( x \) and \( y \).

1. Start with \( y=(x + 8)^2 \)
2. Take square roots: \( \pm\sqrt{y}=x + 8 \)
3. Solve for \( x \): \( x=\pm\sqrt{y}-8 \)
4. Now, swap \( x \) and \( y \) to get the inverse function: \( y=\pm\sqrt{x}-8 \). But the given format is "Inverse of \( d(x)=\pm \) square root [ ] + [ ]". Wait, maybe the format is a bit different. Let's rearrange \( y=\pm\sqrt{x}-8 \) as \( y=\pm\sqrt{x}+(- 8) \). So comparing with the given format "±square root [ ] + [ ]", the first box (inside the square root) should be \( x \), and the second box should be \( (-8) \)? Wait, no, maybe the original function is \( d(x)=(x + 8)^2 \), so when we find the inverse, we have \( x=(y + 8)^2 \), then \( y + 8=\pm\sqrt{x} \), so \( y=\pm\sqrt{x}-8 \). But the given structure is "±square root [ ] + [ ]", so maybe it's a typo or a different way of writing. Wait, if we write \( y=\pm\sqrt{x}+(-8) \), then the first box is \( x \) and the second box is \( (-8) \)? Wait, no, maybe the intended form is \( \pm\sqrt{x}+(-8) \), but the second box is - 8? Wait, the problem says "Inverse of \( d(x)=\pm \) square root [ ] + [ ]". Let's check again.

Let \( y=(x + 8)^2 \). Let's solve for \( x \):

\( y=(x + 8)^2 \)

Take square roots: \( \sqrt{y}=x + 8 \) or \( -\sqrt{y}=x + 8 \)

So \( x=\sqrt{y}-8 \) or \( x =-\sqrt{y}-8 \), which can be written as \( x=\pm\sqrt{y}-8=\pm\sqrt{y}+(-8) \)

When we write the inverse function (let's call it \( d^{-1}(x) \)), we have \( d^{-1}(x)=\pm\sqrt{x}-8=\pm\sqrt{x}+(-8) \)

So in the given format "Inverse of \( d(x)=\pm \) square root [ ] + [ ]", the first box (inside the square root) is \( x \), and the second box is \( (-8) \)? Wait, but maybe the problem has a different way. Wait, maybe the original function is \( d(x)=(x + 8)^2 \), so to find the inverse, we can think of it as:

Let \( y=(x + 8)^2 \), then \( x + 8=\pm\sqrt{y} \), so \( x=\pm\sqrt{y}-8 \). So when we write the inverse function in terms of \( x \), it's \( d^{-1}(x)=\pm\sqrt{x}-8 \). So the "±square root" of \( x \) plus \( (-8) \). So the first box is \( x \) and the second box is \( (-8) \)? Wait, but the second box is a plus sign. Wait, maybe the problem has a typo, and it's supposed to be "±square root [ ] - [ ]", but given the format, we have to fit. So if we write \( \pm\sqrt{x}+(-8) \), then the first box is \( x \) and the second box is \( (-8) \).

Answer:

The first box (inside the square root) is \( x \) and the second box is \( - 8 \). So the first blank is \( x \), the second blank is \( -8 \).